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

Percentage Accurate: 99.9% → 99.9%

Time: 3.9s

Alternatives: 14

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

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

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]

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]
2. Add Preprocessing

Alternative 2: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sin y \cdot z\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* (sin y) z))))
   (if (<= x -1.0) t_0 (if (<= x 5.5e-5) (- (cos y) (* z (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x - (sin(y) * z);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 5.5e-5) {
		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 = x - (sin(y) * z)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 5.5d-5) 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 = x - (Math.sin(y) * z);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 5.5e-5) {
		tmp = Math.cos(y) - (z * Math.sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (math.sin(y) * z)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 5.5e-5:
		tmp = math.cos(y) - (z * math.sin(y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(sin(y) * z))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 5.5e-5)
		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 = x - (sin(y) * z);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 5.5e-5)
		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[(x - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 5.5e-5], 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 < -1 or 5.5000000000000002e-5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 98.1

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

   6. Applied rewrites 98.1%

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

   if -1 < x < 5.5000000000000002e-5

   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 98.7

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

   4. Applied rewrites 98.7%

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

Alternative 3: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - z \cdot \sin y\\ t_1 := x - \sin y \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* z (sin y)))) (t_1 (- x (* (sin y) z))))
   (if (<= t_0 -5e+37) t_1 (if (<= t_0 2e+15) (+ (cos y) x) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (z * sin(y));
	double t_1 = x - (sin(y) * z);
	double tmp;
	if (t_0 <= -5e+37) {
		tmp = t_1;
	} else if (t_0 <= 2e+15) {
		tmp = cos(y) + x;
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_0 = (x + cos(y)) - (z * sin(y))
    t_1 = x - (sin(y) * z)
    if (t_0 <= (-5d+37)) then
        tmp = t_1
    else if (t_0 <= 2d+15) then
        tmp = cos(y) + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + Math.cos(y)) - (z * Math.sin(y));
	double t_1 = x - (Math.sin(y) * z);
	double tmp;
	if (t_0 <= -5e+37) {
		tmp = t_1;
	} else if (t_0 <= 2e+15) {
		tmp = Math.cos(y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.cos(y)) - (z * math.sin(y))
	t_1 = x - (math.sin(y) * z)
	tmp = 0
	if t_0 <= -5e+37:
		tmp = t_1
	elif t_0 <= 2e+15:
		tmp = math.cos(y) + x
	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 - Float64(sin(y) * z))
	tmp = 0.0
	if (t_0 <= -5e+37)
		tmp = t_1;
	elseif (t_0 <= 2e+15)
		tmp = Float64(cos(y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + cos(y)) - (z * sin(y));
	t_1 = x - (sin(y) * z);
	tmp = 0.0;
	if (t_0 <= -5e+37)
		tmp = t_1;
	elseif (t_0 <= 2e+15)
		tmp = cos(y) + x;
	else
		tmp = t_1;
	end
	tmp_2 = 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[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+37], t$95$1, If[LessEqual[t$95$0, 2e+15], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -4.99999999999999989e37 or 2e15 < (-.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 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if -4.99999999999999989e37 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 2e15

   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 93.0

         \[\leadsto \cos y + x \]

   4. Applied rewrites 93.0%

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

Alternative 4: 81.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{+68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+94}:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))))
   (if (<= z -7.6e+68) t_0 (if (<= z 3e+94) (+ (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double tmp;
	if (z <= -7.6e+68) {
		tmp = t_0;
	} else if (z <= 3e+94) {
		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 = -z * sin(y)
    if (z <= (-7.6d+68)) then
        tmp = t_0
    else if (z <= 3d+94) 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 = -z * Math.sin(y);
	double tmp;
	if (z <= -7.6e+68) {
		tmp = t_0;
	} else if (z <= 3e+94) {
		tmp = Math.cos(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * math.sin(y)
	tmp = 0
	if z <= -7.6e+68:
		tmp = t_0
	elif z <= 3e+94:
		tmp = math.cos(y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * sin(y))
	tmp = 0.0
	if (z <= -7.6e+68)
		tmp = t_0;
	elseif (z <= 3e+94)
		tmp = Float64(cos(y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * sin(y);
	tmp = 0.0;
	if (z <= -7.6e+68)
		tmp = t_0;
	elseif (z <= 3e+94)
		tmp = cos(y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.6e+68], t$95$0, If[LessEqual[z, 3e+94], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.6000000000000002e68 or 3.0000000000000001e94 < 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 63.0

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

   4. Applied rewrites 63.0%

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

   if -7.6000000000000002e68 < z < 3.0000000000000001e94

   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 92.6

         \[\leadsto \cos y + x \]

   4. Applied rewrites 92.6%

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

Alternative 5: 80.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y + x\\ \mathbf{if}\;y \leq -20000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 57:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot y\right) \cdot 0.16666666666666666\right) \cdot y - z, y, x\right) - -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (cos y) x)))
   (if (<= y -20000000000.0)
     t_0
     (if (<= y 57.0)
       (- (fma (- (* (* (* z y) 0.16666666666666666) 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 <= -20000000000.0) {
		tmp = t_0;
	} else if (y <= 57.0) {
		tmp = fma(((((z * y) * 0.16666666666666666) * 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 <= -20000000000.0)
		tmp = t_0;
	elseif (y <= 57.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(z * y) * 0.16666666666666666) * y) - z), y, 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, -20000000000.0], t$95$0, If[LessEqual[y, 57.0], N[(N[(N[(N[(N[(N[(z * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision] - -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2e10 or 57 < 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 62.4

         \[\leadsto \cos y + x \]

   4. Applied rewrites 62.4%

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

   if -2e10 < y < 57

   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. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

Alternative 6: 73.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - z \cdot \sin y\\ \mathbf{if}\;t\_0 \leq -50000000000:\\ \;\;\;\;x - z \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.999:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x\right) - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* z (sin y)))))
   (if (<= t_0 -50000000000.0)
     (- x (* z y))
     (if (<= t_0 0.999) (cos y) (- (fma (- y) z x) -1.0)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (z * sin(y));
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = x - (z * y);
	} else if (t_0 <= 0.999) {
		tmp = cos(y);
	} else {
		tmp = fma(-y, z, x) - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
	tmp = 0.0
	if (t_0 <= -50000000000.0)
		tmp = Float64(x - Float64(z * y));
	elseif (t_0 <= 0.999)
		tmp = cos(y);
	else
		tmp = Float64(fma(Float64(-y), z, x) - -1.0);
	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]}, If[LessEqual[t$95$0, -50000000000.0], N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.999], N[Cos[y], $MachinePrecision], N[(N[((-y) * z + x), $MachinePrecision] - -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -5e10

   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 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 65.5%

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

   if -5e10 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.998999999999999999

   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 94.3

         \[\leadsto \cos y + x \]

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \cos y \]

      2. lift-cos.f64 86.7

         \[\leadsto \cos y \]

   7. Applied rewrites 86.7%

      \[\leadsto \cos y \]

   if 0.998999999999999999 < (-.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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.6

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

   4. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 75.6

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

   6. Applied rewrites 75.6%

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

Alternative 7: 69.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+43}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+63}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+43)
   (- x -1.0)
   (if (<= y 1.9e+63)
     (-
      (+ 1.0 x)
      (*
       z
       (*
        (fma
         (fma 0.008333333333333333 (* y y) -0.16666666666666666)
         (* y y)
         1.0)
        y)))
     (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+43) {
		tmp = x - -1.0;
	} else if (y <= 1.9e+63) {
		tmp = (1.0 + x) - (z * (fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * y));
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e+43)
		tmp = Float64(x - -1.0);
	elseif (y <= 1.9e+63)
		tmp = Float64(Float64(1.0 + x) - Float64(z * Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * y)));
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.1e+43], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 1.9e+63], N[(N[(1.0 + x), $MachinePrecision] - N[(z * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.10000000000000002e43 or 1.9000000000000001e63 < 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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 40.2

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

   4. Applied rewrites 40.2%

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

   if -2.10000000000000002e43 < y < 1.9000000000000001e63

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto x - z \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{y}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x - z \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{y}\right) \]

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. negate-sub N/A

         \[\leadsto x - z \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {y}^{2}, 1\right) \cdot y\right) \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.8

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

   7. Applied rewrites 57.8%

      \[\leadsto x - z \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(1 + x\right)} - z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), y \cdot y, 1\right) \cdot y\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 89.0

         \[\leadsto \left(1 + \color{blue}{x}\right) - z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \]

   10. Applied rewrites 89.0%

       \[\leadsto \color{blue}{\left(1 + x\right)} - z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+44}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 1600000:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot y\right) \cdot 0.16666666666666666\right) \cdot y - z, y, x\right) - -1\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e+44)
   (- x -1.0)
   (if (<= y 1600000.0)
     (- (fma (- (* (* (* z y) 0.16666666666666666) y) z) y x) -1.0)
     (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e+44) {
		tmp = x - -1.0;
	} else if (y <= 1600000.0) {
		tmp = fma(((((z * y) * 0.16666666666666666) * y) - z), y, x) - -1.0;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.7e+44)
		tmp = Float64(x - -1.0);
	elseif (y <= 1600000.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(z * y) * 0.16666666666666666) * y) - z), y, x) - -1.0);
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.7e+44], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 1600000.0], N[(N[(N[(N[(N[(N[(z * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision] - -1.0), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e44 or 1.6e6 < 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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 40.3

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

   4. Applied rewrites 40.3%

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

   if -2.7e44 < y < 1.6e6

   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. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 94.8

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

   7. Applied rewrites 94.8%

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

Alternative 9: 69.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+19}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x\right) - -1\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.45e+19)
   (- x -1.0)
   (if (<= y 3.05e+82) (- (fma (- y) z x) -1.0) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.45e+19) {
		tmp = x - -1.0;
	} else if (y <= 3.05e+82) {
		tmp = fma(-y, z, x) - -1.0;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.45e+19)
		tmp = Float64(x - -1.0);
	elseif (y <= 3.05e+82)
		tmp = Float64(fma(Float64(-y), z, x) - -1.0);
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.45e+19], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 3.05e+82], N[(N[((-y) * z + x), $MachinePrecision] - -1.0), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e19 or 3.0499999999999999e82 < 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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 40.2

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

   4. Applied rewrites 40.2%

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

   if -1.45e19 < y < 3.0499999999999999e82

   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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. negate-sub N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 90.7

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

   6. Applied rewrites 90.7%

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

Alternative 10: 68.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+19}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{+82}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.45e+19)
   (- x -1.0)
   (if (<= y 3.05e+82) (- x (fma z y -1.0)) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.45e+19) {
		tmp = x - -1.0;
	} else if (y <= 3.05e+82) {
		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 <= -1.45e+19)
		tmp = Float64(x - -1.0);
	elseif (y <= 3.05e+82)
		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, -1.45e+19], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 3.05e+82], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e19 or 3.0499999999999999e82 < 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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 40.2

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

   4. Applied rewrites 40.2%

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

   if -1.45e19 < y < 3.0499999999999999e82

   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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.7

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\left(x - z \cdot y\right) - -1} \]
   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 90.7

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

   6. Applied rewrites 90.7%

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

Alternative 11: 66.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -96000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;1 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -96000.0) x (if (<= x 1.9e-8) (- 1.0 (* z y)) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -96000.0) {
		tmp = x;
	} else if (x <= 1.9e-8) {
		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 <= (-96000.0d0)) then
        tmp = x
    else if (x <= 1.9d-8) 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 <= -96000.0) {
		tmp = x;
	} else if (x <= 1.9e-8) {
		tmp = 1.0 - (z * y);
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -96000.0:
		tmp = x
	elif x <= 1.9e-8:
		tmp = 1.0 - (z * y)
	else:
		tmp = x - -1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -96000.0)
		tmp = x;
	elseif (x <= 1.9e-8)
		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 <= -96000.0)
		tmp = x;
	elseif (x <= 1.9e-8)
		tmp = 1.0 - (z * y);
	else
		tmp = x - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -96000.0], x, If[LessEqual[x, 1.9e-8], N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -96000

   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 82.6%

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

   if -96000 < x < 1.90000000000000014e-8

   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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.3

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

   4. Applied rewrites 52.3%

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

   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 51.3

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

   7. Applied rewrites 51.3%

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

   if 1.90000000000000014e-8 < 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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 79.2

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

   4. Applied rewrites 79.2%

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

Alternative 12: 61.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{+205}:\\ \;\;\;\;x - -1\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.4e+205) (- x -1.0) (* (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.4e+205) {
		tmp = x - -1.0;
	} else {
		tmp = -y * z;
	}
	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 (z <= 1.4d+205) then
        tmp = x - (-1.0d0)
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.4e+205) {
		tmp = x - -1.0;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.4e+205:
		tmp = x - -1.0
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.4e+205)
		tmp = Float64(x - -1.0);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.4e+205)
		tmp = x - -1.0;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.4e+205], N[(x - -1.0), $MachinePrecision], N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.39999999999999996e205

   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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 64.5

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

   4. Applied rewrites 64.5%

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

   if 1.39999999999999996e205 < z

   1. Initial program 99.8%

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

   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

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 53.7

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

   4. Applied rewrites 53.7%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 33.7

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

   7. Applied rewrites 33.7%

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

Alternative 13: 60.9% 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

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

   3. negate-sub-reverse N/A

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

   4. lower--.f64 60.9

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

4. Applied rewrites 60.9%

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

Alternative 14: 42.0% 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 42.0%

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

Reproduce

herbie shell --seed 2025116 
(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))))