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

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

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
3. Add Preprocessing

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + \cos y\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -2000000000 \lor \neg \left(t\_1 \leq 0.998\right):\\ \;\;\;\;\left(x + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos y}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- (+ x (cos y)) t_0)))
   (if (or (<= t_1 -2000000000.0) (not (<= t_1 0.998)))
     (- (+ x 1.0) t_0)
     (* (/ (cos y) x) x))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = (x + cos(y)) - t_0;
	double tmp;
	if ((t_1 <= -2000000000.0) || !(t_1 <= 0.998)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = (cos(y) / x) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = (x + cos(y)) - t_0
    if ((t_1 <= (-2000000000.0d0)) .or. (.not. (t_1 <= 0.998d0))) then
        tmp = (x + 1.0d0) - t_0
    else
        tmp = (cos(y) / x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double t_1 = (x + Math.cos(y)) - t_0;
	double tmp;
	if ((t_1 <= -2000000000.0) || !(t_1 <= 0.998)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = (Math.cos(y) / x) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = (x + math.cos(y)) - t_0
	tmp = 0
	if (t_1 <= -2000000000.0) or not (t_1 <= 0.998):
		tmp = (x + 1.0) - t_0
	else:
		tmp = (math.cos(y) / x) * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(Float64(x + cos(y)) - t_0)
	tmp = 0.0
	if ((t_1 <= -2000000000.0) || !(t_1 <= 0.998))
		tmp = Float64(Float64(x + 1.0) - t_0);
	else
		tmp = Float64(Float64(cos(y) / x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = (x + cos(y)) - t_0;
	tmp = 0.0;
	if ((t_1 <= -2000000000.0) || ~((t_1 <= 0.998)))
		tmp = (x + 1.0) - t_0;
	else
		tmp = (cos(y) / x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2000000000.0], N[Not[LessEqual[t$95$1, 0.998]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[Cos[y], $MachinePrecision] / x), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -2e9 or 0.998 < (-.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. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.0%

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

   if -2e9 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.998

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in z around 0

      \[\leadsto \left(1 + \frac{\cos y}{x}\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 99.7%

       \[\leadsto \frac{\cos y + x}{x} \cdot x \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{\cos y}{x} \cdot x \]
   13. Step-by-step derivation

   14. Applied rewrites 96.5%

       \[\leadsto \frac{\cos y}{x} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -2000000000 \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 0.998\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos y}{x} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+26} \lor \neg \left(t\_2 \leq 0.99999999995\right):\\ \;\;\;\;\left(x + 1\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))) (t_1 (* z (sin y))) (t_2 (- t_0 t_1)))
   (if (or (<= t_2 -5e+26) (not (<= t_2 0.99999999995)))
     (- (+ x 1.0) t_1)
     (- t_0 (* z y)))))

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 tmp;
	if ((t_2 <= -5e+26) || !(t_2 <= 0.99999999995)) {
		tmp = (x + 1.0) - t_1;
	} else {
		tmp = t_0 - (z * y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x + cos(y)
    t_1 = z * sin(y)
    t_2 = t_0 - t_1
    if ((t_2 <= (-5d+26)) .or. (.not. (t_2 <= 0.99999999995d0))) then
        tmp = (x + 1.0d0) - t_1
    else
        tmp = t_0 - (z * y)
    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 tmp;
	if ((t_2 <= -5e+26) || !(t_2 <= 0.99999999995)) {
		tmp = (x + 1.0) - t_1;
	} else {
		tmp = t_0 - (z * y);
	}
	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
	tmp = 0
	if (t_2 <= -5e+26) or not (t_2 <= 0.99999999995):
		tmp = (x + 1.0) - t_1
	else:
		tmp = t_0 - (z * y)
	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)
	tmp = 0.0
	if ((t_2 <= -5e+26) || !(t_2 <= 0.99999999995))
		tmp = Float64(Float64(x + 1.0) - t_1);
	else
		tmp = Float64(t_0 - Float64(z * y));
	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;
	tmp = 0.0;
	if ((t_2 <= -5e+26) || ~((t_2 <= 0.99999999995)))
		tmp = (x + 1.0) - t_1;
	else
		tmp = t_0 - (z * y);
	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]}, If[Or[LessEqual[t$95$2, -5e+26], N[Not[LessEqual[t$95$2, 0.99999999995]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - N[(z * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -5.0000000000000001e26 or 0.99999999995 < (-.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. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

   if -5.0000000000000001e26 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.99999999995

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -5 \cdot 10^{+26} \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 0.99999999995\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-6} \lor \neg \left(z \leq 1.05 \cdot 10^{-27}\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos y + x}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.9e-6) (not (<= z 1.05e-27)))
   (- (+ x 1.0) (* z (sin y)))
   (* (/ (+ (cos y) x) x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.9e-6) || !(z <= 1.05e-27)) {
		tmp = (x + 1.0) - (z * sin(y));
	} else {
		tmp = ((cos(y) + x) / x) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.9d-6)) .or. (.not. (z <= 1.05d-27))) then
        tmp = (x + 1.0d0) - (z * sin(y))
    else
        tmp = ((cos(y) + x) / x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.9e-6) || !(z <= 1.05e-27)) {
		tmp = (x + 1.0) - (z * Math.sin(y));
	} else {
		tmp = ((Math.cos(y) + x) / x) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.9e-6) or not (z <= 1.05e-27):
		tmp = (x + 1.0) - (z * math.sin(y))
	else:
		tmp = ((math.cos(y) + x) / x) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.9e-6) || !(z <= 1.05e-27))
		tmp = Float64(Float64(x + 1.0) - Float64(z * sin(y)));
	else
		tmp = Float64(Float64(Float64(cos(y) + x) / x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.9e-6) || ~((z <= 1.05e-27)))
		tmp = (x + 1.0) - (z * sin(y));
	else
		tmp = ((cos(y) + x) / x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.9e-6], N[Not[LessEqual[z, 1.05e-27]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision] / x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9e-6 or 1.05000000000000008e-27 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.8%

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

   if -1.9e-6 < z < 1.05000000000000008e-27

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in z around 0

      \[\leadsto \left(1 + \frac{\cos y}{x}\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

       \[\leadsto \frac{\cos y + x}{x} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-6} \lor \neg \left(z \leq 1.05 \cdot 10^{-27}\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos y + x}{x} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+161}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+29}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+16}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.85e+161)
   (+ 1.0 x)
   (if (<= y -1e+29)
     (* (- z) (sin y))
     (if (<= y 2.95e+16) (- x (fma z y -1.0)) (+ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.85e+161) {
		tmp = 1.0 + x;
	} else if (y <= -1e+29) {
		tmp = -z * sin(y);
	} else if (y <= 2.95e+16) {
		tmp = x - fma(z, y, -1.0);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.85e+161)
		tmp = Float64(1.0 + x);
	elseif (y <= -1e+29)
		tmp = Float64(Float64(-z) * sin(y));
	elseif (y <= 2.95e+16)
		tmp = Float64(x - fma(z, y, -1.0));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.85e+161], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, -1e+29], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.95e+16], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8499999999999999e161 or 2.95e16 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 49.8

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

   5. Applied rewrites 49.8%

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

   if -1.8499999999999999e161 < y < -9.99999999999999914e28

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 53.1

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

   5. Applied rewrites 53.1%

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

   if -9.99999999999999914e28 < y < 2.95e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. fp-cancel-sub-sign N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 95.6

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

   8. Applied rewrites 95.6%

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

Alternative 6: 88.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(x + 1\right) - z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ x 1.0) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x + 1.0) - (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 + 1.0d0) - (z * sin(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + 1.0) - (z * Math.sin(y));
}

Python

def code(x, y, z):
	return (x + 1.0) - (z * math.sin(y))

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 85.6%

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

Alternative 7: 69.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4200 \lor \neg \left(y \leq 23\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4200.0) (not (<= y 23.0)))
   (+ 1.0 x)
   (-
    (+ x (fma (* y y) -0.5 1.0))
    (*
     (fma
      (* z (fma 0.008333333333333333 (* y y) -0.16666666666666666))
      (* y y)
      z)
     y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4200.0) || !(y <= 23.0)) {
		tmp = 1.0 + x;
	} else {
		tmp = (x + fma((y * y), -0.5, 1.0)) - (fma((z * fma(0.008333333333333333, (y * y), -0.16666666666666666)), (y * y), z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4200.0) || !(y <= 23.0))
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(Float64(x + fma(Float64(y * y), -0.5, 1.0)) - Float64(fma(Float64(z * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666)), Float64(y * y), z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4200.0], N[Not[LessEqual[y, 23.0]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[(x + N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(z * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4200 or 23 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 43.4

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

   5. Applied rewrites 43.4%

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

   if -4200 < y < 23

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.5

          \[\leadsto \left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), \color{blue}{y \cdot y}, z\right) \cdot y \]

   8. Applied rewrites 99.5%

      \[\leadsto \left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \color{blue}{\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4200 \lor \neg \left(y \leq 23\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\right) - \mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4200 \lor \neg \left(y \leq 30\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4200.0) (not (<= y 30.0)))
   (+ 1.0 x)
   (fma (- (* (- (* 0.16666666666666666 (* z y)) 0.5) y) z) y (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4200.0) || !(y <= 30.0)) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(((((0.16666666666666666 * (z * y)) - 0.5) * y) - z), y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4200.0) || !(y <= 30.0))
		tmp = Float64(1.0 + x);
	else
		tmp = fma(Float64(Float64(Float64(Float64(0.16666666666666666 * Float64(z * y)) - 0.5) * y) - z), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4200.0], N[Not[LessEqual[y, 30.0]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4200 or 30 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 43.4

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

   5. Applied rewrites 43.4%

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

   if -4200 < y < 30

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4200 \lor \neg \left(y \leq 30\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4500:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 5000000000:\\ \;\;\;\;x - \mathsf{fma}\left(\mathsf{fma}\left(0.5, y, z\right), y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -1}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4500.0)
   (+ 1.0 x)
   (if (<= y 5000000000.0)
     (- x (fma (fma 0.5 y z) y -1.0))
     (* (/ (- x -1.0) x) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4500.0) {
		tmp = 1.0 + x;
	} else if (y <= 5000000000.0) {
		tmp = x - fma(fma(0.5, y, z), y, -1.0);
	} else {
		tmp = ((x - -1.0) / x) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4500.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 5000000000.0)
		tmp = Float64(x - fma(fma(0.5, y, z), y, -1.0));
	else
		tmp = Float64(Float64(Float64(x - -1.0) / x) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4500.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 5000000000.0], N[(x - N[(N[(0.5 * y + z), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - -1.0), $MachinePrecision] / x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4500

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 45.8

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

   5. Applied rewrites 45.8%

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

   if -4500 < y < 5e9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-+l- N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

   8. Applied rewrites 98.7%

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

   if 5e9 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 90.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 41.2%

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

Alternative 10: 69.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4500 \lor \neg \left(y \leq 5000000000\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\mathsf{fma}\left(0.5, y, z\right), y, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4500.0) (not (<= y 5000000000.0)))
   (+ 1.0 x)
   (- x (fma (fma 0.5 y z) y -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4500.0) || !(y <= 5000000000.0)) {
		tmp = 1.0 + x;
	} else {
		tmp = x - fma(fma(0.5, y, z), y, -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4500.0) || !(y <= 5000000000.0))
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(x - fma(fma(0.5, y, z), y, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4500.0], N[Not[LessEqual[y, 5000000000.0]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(x - N[(N[(0.5 * y + z), $MachinePrecision] * y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4500 or 5e9 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 43.6

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

   5. Applied rewrites 43.6%

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

   if -4500 < y < 5e9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-+l- N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

   8. Applied rewrites 98.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4500 \lor \neg \left(y \leq 5000000000\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\mathsf{fma}\left(0.5, y, z\right), y, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+21} \lor \neg \left(y \leq 1000000000\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.5e+21) (not (<= y 1000000000.0)))
   (+ 1.0 x)
   (- x (fma z y -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.5e+21) || !(y <= 1000000000.0)) {
		tmp = 1.0 + x;
	} else {
		tmp = x - fma(z, y, -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.5e+21) || !(y <= 1000000000.0))
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(x - fma(z, y, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.5e+21], N[Not[LessEqual[y, 1000000000.0]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e21 or 1e9 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 42.1

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

   5. Applied rewrites 42.1%

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

   if -4.5e21 < y < 1e9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. fp-cancel-sub-sign N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 97.3

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

   8. Applied rewrites 97.3%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+21} \lor \neg \left(y \leq 1000000000\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+219} \lor \neg \left(z \leq 7.2 \cdot 10^{+189}\right):\\ \;\;\;\;x - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.5e+219) (not (<= z 7.2e+189))) (- x (* z y)) (+ 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.5e+219) || !(z <= 7.2e+189)) {
		tmp = x - (z * y);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-4.5d+219)) .or. (.not. (z <= 7.2d+189))) then
        tmp = x - (z * y)
    else
        tmp = 1.0d0 + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.5e+219) || !(z <= 7.2e+189)) {
		tmp = x - (z * y);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.5e+219) or not (z <= 7.2e+189):
		tmp = x - (z * y)
	else:
		tmp = 1.0 + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.5e+219) || !(z <= 7.2e+189))
		tmp = Float64(x - Float64(z * y));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.5e+219) || ~((z <= 7.2e+189)))
		tmp = x - (z * y);
	else
		tmp = 1.0 + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.5e+219], N[Not[LessEqual[z, 7.2e+189]], $MachinePrecision]], N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.50000000000000023e219 or 7.20000000000000017e189 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. fp-cancel-sub-sign N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 58.2

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

   8. Applied rewrites 58.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 58.2%

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

   if -4.50000000000000023e219 < z < 7.20000000000000017e189

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 69.4

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

   5. Applied rewrites 69.4%

      \[\leadsto \color{blue}{1 + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+219} \lor \neg \left(z \leq 7.2 \cdot 10^{+189}\right):\\ \;\;\;\;x - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-11} \lor \neg \left(x \leq 1.12 \cdot 10^{-119}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;1 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.8e-11) (not (<= x 1.12e-119))) (+ 1.0 x) (- 1.0 (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.8e-11) || !(x <= 1.12e-119)) {
		tmp = 1.0 + x;
	} else {
		tmp = 1.0 - (z * y);
	}
	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 <= (-7.8d-11)) .or. (.not. (x <= 1.12d-119))) then
        tmp = 1.0d0 + x
    else
        tmp = 1.0d0 - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.8e-11) || !(x <= 1.12e-119)) {
		tmp = 1.0 + x;
	} else {
		tmp = 1.0 - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.8e-11) or not (x <= 1.12e-119):
		tmp = 1.0 + x
	else:
		tmp = 1.0 - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.8e-11) || !(x <= 1.12e-119))
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(1.0 - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.8e-11) || ~((x <= 1.12e-119)))
		tmp = 1.0 + x;
	else
		tmp = 1.0 - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.8e-11], N[Not[LessEqual[x, 1.12e-119]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.80000000000000021e-11 or 1.11999999999999998e-119 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if -7.80000000000000021e-11 < x < 1.11999999999999998e-119

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. fp-cancel-sub-sign N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 54.7

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

   8. Applied rewrites 54.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 54.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-11} \lor \neg \left(x \leq 1.12 \cdot 10^{-119}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;1 - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.8% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+234}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.85e+234) (* (- y) z) (+ 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.85e+234) {
		tmp = -y * z;
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.85d+234)) then
        tmp = -y * z
    else
        tmp = 1.0d0 + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.85e+234) {
		tmp = -y * z;
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.85e+234:
		tmp = -y * z
	else:
		tmp = 1.0 + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.85e+234)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.85e+234)
		tmp = -y * z;
	else
		tmp = 1.0 + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.85e+234], N[((-y) * z), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.84999999999999983e234

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot y\right) \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 47.2%

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

   if -1.84999999999999983e234 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 65.1

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

   5. Applied rewrites 65.1%

      \[\leadsto \color{blue}{1 + x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.1% accurate, 53.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 1.0 + 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 = 1.0d0 + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. lower-+.f64 62.9

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

5. Applied rewrites 62.9%

   \[\leadsto \color{blue}{1 + x} \]
6. Add Preprocessing

Alternative 16: 21.2% accurate, 212.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. lower-+.f64 62.9

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

5. Applied rewrites 62.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 21.9%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025015 
(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))))