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

Percentage Accurate: 99.9% → 99.9%

Time: 4.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

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

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

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. add-flip N/A

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

   5. sub-negate-rev N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.9%

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.9%

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

3. Applied rewrites 99.9%

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

Alternative 2: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left(x + 1\right) - z \cdot \sin y\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (- (+ x 1.0) (* z (sin y)))))
  (if (<= z -1.7e+31) t_0 (if (<= z 1.05e-10) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + 1.0) - (z * sin(y));
	double tmp;
	if (z <= -1.7e+31) {
		tmp = t_0;
	} else if (z <= 1.05e-10) {
		tmp = x + cos(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 = (x + 1.0d0) - (z * sin(y))
    if (z <= (-1.7d+31)) then
        tmp = t_0
    else if (z <= 1.05d-10) then
        tmp = x + cos(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 = (x + 1.0) - (z * Math.sin(y));
	double tmp;
	if (z <= -1.7e+31) {
		tmp = t_0;
	} else if (z <= 1.05e-10) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + 1.0) - (z * math.sin(y))
	tmp = 0
	if z <= -1.7e+31:
		tmp = t_0
	elif z <= 1.05e-10:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + 1.0) - Float64(z * sin(y)))
	tmp = 0.0
	if (z <= -1.7e+31)
		tmp = t_0;
	elseif (z <= 1.05e-10)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + 1.0) - (z * sin(y));
	tmp = 0.0;
	if (z <= -1.7e+31)
		tmp = t_0;
	elseif (z <= 1.05e-10)
		tmp = x + cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.7e+31], t$95$0, If[LessEqual[z, 1.05e-10], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6999999999999999e31 or 1.05e-10 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 88.1%

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

   if -1.6999999999999999e31 < z < 1.05e-10

   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. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 99.9%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 99.8%

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in z around 0

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

       1. lower-+.f64 N/A

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

       2. lower-cos.f64 73.8%

          \[\leadsto x + \cos y \]

   12. Applied rewrites 73.8%

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

Alternative 3: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ t_3 := x - t\_1\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 500000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (+ x (cos y)))
       (t_1 (* z (sin y)))
       (t_2 (- t_0 t_1))
       (t_3 (- x t_1)))
  (if (<= t_2 -2e+19) t_3 (if (<= t_2 500000000.0) t_0 t_3))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double t_1 = z * sin(y);
	double t_2 = t_0 - t_1;
	double t_3 = x - t_1;
	double tmp;
	if (t_2 <= -2e+19) {
		tmp = t_3;
	} else if (t_2 <= 500000000.0) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = x + cos(y)
    t_1 = z * sin(y)
    t_2 = t_0 - t_1
    t_3 = x - t_1
    if (t_2 <= (-2d+19)) then
        tmp = t_3
    else if (t_2 <= 500000000.0d0) then
        tmp = t_0
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + Math.cos(y);
	double t_1 = z * Math.sin(y);
	double t_2 = t_0 - t_1;
	double t_3 = x - t_1;
	double tmp;
	if (t_2 <= -2e+19) {
		tmp = t_3;
	} else if (t_2 <= 500000000.0) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + math.cos(y)
	t_1 = z * math.sin(y)
	t_2 = t_0 - t_1
	t_3 = x - t_1
	tmp = 0
	if t_2 <= -2e+19:
		tmp = t_3
	elif t_2 <= 500000000.0:
		tmp = t_0
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	t_1 = Float64(z * sin(y))
	t_2 = Float64(t_0 - t_1)
	t_3 = Float64(x - t_1)
	tmp = 0.0
	if (t_2 <= -2e+19)
		tmp = t_3;
	elseif (t_2 <= 500000000.0)
		tmp = t_0;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + cos(y);
	t_1 = z * sin(y);
	t_2 = t_0 - t_1;
	t_3 = x - t_1;
	tmp = 0.0;
	if (t_2 <= -2e+19)
		tmp = t_3;
	elseif (t_2 <= 500000000.0)
		tmp = t_0;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+19], t$95$3, If[LessEqual[t$95$2, 500000000.0], t$95$0, t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -2e19 or 5e8 < (-.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. 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. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 99.9%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 99.8%

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 68.5%

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

   12. Applied rewrites 68.5%

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

   if -2e19 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 5e8

   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. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 99.9%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 99.8%

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in z around 0

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

       1. lower-+.f64 N/A

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

       2. lower-cos.f64 73.8%

          \[\leadsto x + \cos y \]

   12. Applied rewrites 73.8%

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

Alternative 4: 81.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := x + \cos y\\ \mathbf{if}\;y \leq -0.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot z, y, -0.5\right) \cdot y, y, x\right) - \left(z \cdot y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (let* ((t_0 (+ x (cos y))))
  (if (<= y -0.35)
    t_0
    (if (<= y 7e-14)
      (-
       (fma (* (fma (* 0.16666666666666666 z) y -0.5) y) y x)
       (- (* z y) 1.0))
      t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double tmp;
	if (y <= -0.35) {
		tmp = t_0;
	} else if (y <= 7e-14) {
		tmp = fma((fma((0.16666666666666666 * z), y, -0.5) * y), y, x) - ((z * y) - 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	tmp = 0.0
	if (y <= -0.35)
		tmp = t_0;
	elseif (y <= 7e-14)
		tmp = Float64(fma(Float64(fma(Float64(0.16666666666666666 * z), y, -0.5) * y), y, x) - Float64(Float64(z * y) - 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.35], t$95$0, If[LessEqual[y, 7e-14], N[(N[(N[(N[(N[(0.16666666666666666 * z), $MachinePrecision] * y + -0.5), $MachinePrecision] * y), $MachinePrecision] * y + x), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.34999999999999998 or 7.0000000000000005e-14 < y

   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. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 99.9%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 99.8%

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in z around 0

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

       1. lower-+.f64 N/A

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

       2. lower-cos.f64 73.8%

          \[\leadsto x + \cos y \]

   12. Applied rewrites 73.8%

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

   if -0.34999999999999998 < y < 7.0000000000000005e-14

   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. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 55.2%

         \[\leadsto 1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right) \]

   4. Applied rewrites 55.2%

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-lft-in N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. add-flip N/A

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

      10. lower--.f64 N/A

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

   6. Applied rewrites 54.8%

      \[\leadsto 1 + \left(x + \left(\mathsf{fma}\left(0.16666666666666666 \cdot z, y, -0.5\right) \cdot \left(y \cdot y\right) - \color{blue}{z \cdot y}\right)\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-+r- N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 54.7%

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

   8. Applied rewrites 54.7%

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

Alternative 5: 70.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -0.32:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot z, y, -0.5\right) \cdot y, y, x\right) - \left(z \cdot y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= y -0.32)
  (- x -1.0)
  (if (<= y 7e-14)
    (-
     (fma (* (fma (* 0.16666666666666666 z) y -0.5) y) y x)
     (- (* z y) 1.0))
    (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.32) {
		tmp = x - -1.0;
	} else if (y <= 7e-14) {
		tmp = fma((fma((0.16666666666666666 * z), y, -0.5) * y), y, x) - ((z * y) - 1.0);
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.32)
		tmp = Float64(x - -1.0);
	elseif (y <= 7e-14)
		tmp = Float64(fma(Float64(fma(Float64(0.16666666666666666 * z), y, -0.5) * y), y, x) - Float64(Float64(z * y) - 1.0));
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.32], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 7e-14], N[(N[(N[(N[(N[(0.16666666666666666 * z), $MachinePrecision] * y + -0.5), $MachinePrecision] * y), $MachinePrecision] * y + x), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.32000000000000001 or 7.0000000000000005e-14 < y

   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. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 99.9%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 62.5%

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

   if -0.32000000000000001 < y < 7.0000000000000005e-14

   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. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 55.2%

         \[\leadsto 1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right) \]

   4. Applied rewrites 55.2%

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-lft-in N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. add-flip N/A

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

      10. lower--.f64 N/A

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

   6. Applied rewrites 54.8%

      \[\leadsto 1 + \left(x + \left(\mathsf{fma}\left(0.16666666666666666 \cdot z, y, -0.5\right) \cdot \left(y \cdot y\right) - \color{blue}{z \cdot y}\right)\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-+r- N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 54.7%

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

   8. Applied rewrites 54.7%

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

Alternative 6: 70.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -0.32:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-14}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= y -0.32)
  (- x -1.0)
  (if (<= y 7e-14)
    (+
     1.0
     (+ x (* y (- (* y (- (* 0.16666666666666666 (* y z)) 0.5)) z))))
    (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.32) {
		tmp = x - -1.0;
	} else if (y <= 7e-14) {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	} 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 (y <= (-0.32d0)) then
        tmp = x - (-1.0d0)
    else if (y <= 7d-14) then
        tmp = 1.0d0 + (x + (y * ((y * ((0.16666666666666666d0 * (y * z)) - 0.5d0)) - z)))
    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 (y <= -0.32) {
		tmp = x - -1.0;
	} else if (y <= 7e-14) {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.32:
		tmp = x - -1.0
	elif y <= 7e-14:
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)))
	else:
		tmp = x - -1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.32)
		tmp = Float64(x - -1.0);
	elseif (y <= 7e-14)
		tmp = Float64(1.0 + Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(0.16666666666666666 * Float64(y * z)) - 0.5)) - z))));
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -0.32)
		tmp = x - -1.0;
	elseif (y <= 7e-14)
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	else
		tmp = x - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.32], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 7e-14], N[(1.0 + N[(x + N[(y * N[(N[(y * N[(N[(0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.32000000000000001 or 7.0000000000000005e-14 < y

   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. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 99.9%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 62.5%

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

   if -0.32000000000000001 < y < 7.0000000000000005e-14

   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. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 55.2%

         \[\leadsto 1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right) \]

   4. Applied rewrites 55.2%

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

Alternative 7: 69.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -0.32:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-14}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= y -0.32)
  (- x -1.0)
  (if (<= y 7e-14) (- x (fma z y -1.0)) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.32) {
		tmp = x - -1.0;
	} else if (y <= 7e-14) {
		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 <= -0.32)
		tmp = Float64(x - -1.0);
	elseif (y <= 7e-14)
		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, -0.32], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 7e-14], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.32000000000000001 or 7.0000000000000005e-14 < y

   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. associate--l+ N/A

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

      4. add-flip N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 99.9%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 62.5%

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

   if -0.32000000000000001 < y < 7.0000000000000005e-14

   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. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.0%

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

   4. Applied rewrites 64.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. add-flip-rev N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 64.0%

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

   6. Applied rewrites 64.0%

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

Alternative 8: 62.5% accurate, 21.2× speedup

Math

\[x - -1 \]

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. 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. associate--l+ N/A

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

   4. add-flip N/A

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

   5. sub-negate-rev N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 99.9%

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.9%

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in y around 0

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

6. Applied rewrites 62.5%

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

Alternative 9: 43.2% accurate, 25.5× speedup

Math

\[-\left(-x\right) \]

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 Float64(-Float64(-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 y around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 64.0%

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

4. Applied rewrites 64.0%

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

5. Taylor expanded in z around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 51.7%

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

7. Applied rewrites 51.7%

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

8. Taylor expanded in x around inf

   \[\leadsto -1 \cdot \left(-1 \cdot x\right) \]
9. Step-by-step derivation

   1. lower-*.f64 43.2%

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

10. Applied rewrites 43.2%

    \[\leadsto -1 \cdot \left(-1 \cdot x\right) \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. mul-1-neg N/A

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

    3. lower-neg.f64 43.2%

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

    4. lift-*.f64 N/A

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

    5. mul-1-neg N/A

       \[\leadsto -\left(\mathsf{neg}\left(x\right)\right) \]

    6. lower-neg.f64 43.2%

       \[\leadsto -\left(-x\right) \]

12. Applied rewrites 43.2%

    \[\leadsto -\left(-x\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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))))