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

Percentage Accurate: 99.9% → 99.9%

Time: 3.8s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, z, x\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.86:\\ \;\;\;\;\left(x + \sin y\right) + z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (cos y) z x)))
   (if (<= z -1.0) t_0 (if (<= z 0.86) (+ (+ x (sin y)) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(cos(y), z, x);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 0.86) {
		tmp = (x + sin(y)) + z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(cos(y), z, x)
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 0.86)
		tmp = Float64(Float64(x + sin(y)) + z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 0.86], N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.859999999999999987 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\cos y, z, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.4%

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

   if -1 < z < 0.859999999999999987

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 99.5%

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

Alternative 3: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y + x\\ t_1 := \left(x + \sin y\right) + z \cdot \cos y\\ t_2 := \mathsf{fma}\left(\cos y, z, x\right)\\ \mathbf{if}\;t\_1 \leq -5000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;\left(z + y\right) + x\\ \mathbf{elif}\;t\_1 \leq 50000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (sin y) x))
        (t_1 (+ (+ x (sin y)) (* z (cos y))))
        (t_2 (fma (cos y) z x)))
   (if (<= t_1 -5000000000.0)
     t_2
     (if (<= t_1 -5e-5)
       t_0
       (if (<= t_1 1e-8) (+ (+ z y) x) (if (<= t_1 50000000000.0) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) + x;
	double t_1 = (x + sin(y)) + (z * cos(y));
	double t_2 = fma(cos(y), z, x);
	double tmp;
	if (t_1 <= -5000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -5e-5) {
		tmp = t_0;
	} else if (t_1 <= 1e-8) {
		tmp = (z + y) + x;
	} else if (t_1 <= 50000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) + x)
	t_1 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	t_2 = fma(cos(y), z, x)
	tmp = 0.0
	if (t_1 <= -5000000000.0)
		tmp = t_2;
	elseif (t_1 <= -5e-5)
		tmp = t_0;
	elseif (t_1 <= 1e-8)
		tmp = Float64(Float64(z + y) + x);
	elseif (t_1 <= 50000000000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[y], $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000.0], t$95$2, If[LessEqual[t$95$1, -5e-5], t$95$0, If[LessEqual[t$95$1, 1e-8], N[(N[(z + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 50000000000.0], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -5e9 or 5e10 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\cos y, z, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.8%

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

   if -5e9 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -5.00000000000000024e-5 or 1e-8 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 5e10

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sin.f64 90.5

         \[\leadsto \sin y + x \]

   4. Applied rewrites 90.5%

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

   if -5.00000000000000024e-5 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 4: 83.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-41}:\\ \;\;\;\;\sin y + x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+196}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -2.35e-7)
     t_0
     (if (<= z 9.2e-41) (+ (sin y) x) (if (<= z 3e+196) (+ z x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -2.35e-7) {
		tmp = t_0;
	} else if (z <= 9.2e-41) {
		tmp = sin(y) + x;
	} else if (z <= 3e+196) {
		tmp = z + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(y) * z
    if (z <= (-2.35d-7)) then
        tmp = t_0
    else if (z <= 9.2d-41) then
        tmp = sin(y) + x
    else if (z <= 3d+196) then
        tmp = z + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * z;
	double tmp;
	if (z <= -2.35e-7) {
		tmp = t_0;
	} else if (z <= 9.2e-41) {
		tmp = Math.sin(y) + x;
	} else if (z <= 3e+196) {
		tmp = z + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * z
	tmp = 0
	if z <= -2.35e-7:
		tmp = t_0
	elif z <= 9.2e-41:
		tmp = math.sin(y) + x
	elif z <= 3e+196:
		tmp = z + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -2.35e-7)
		tmp = t_0;
	elseif (z <= 9.2e-41)
		tmp = Float64(sin(y) + x);
	elseif (z <= 3e+196)
		tmp = Float64(z + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * z;
	tmp = 0.0;
	if (z <= -2.35e-7)
		tmp = t_0;
	elseif (z <= 9.2e-41)
		tmp = sin(y) + x;
	elseif (z <= 3e+196)
		tmp = z + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.35e-7], t$95$0, If[LessEqual[z, 9.2e-41], N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3e+196], N[(z + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.35e-7 or 2.9999999999999999e196 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \cos y \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos y \cdot \color{blue}{z} \]

      3. lift-cos.f64 76.9

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

   4. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   if -2.35e-7 < z < 9.20000000000000041e-41

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sin.f64 92.7

         \[\leadsto \sin y + x \]

   4. Applied rewrites 92.7%

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

   if 9.20000000000000041e-41 < z < 2.9999999999999999e196

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 71.5

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

   4. Applied rewrites 71.5%

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

Alternative 5: 81.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(z + y\right) + x\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (<= t_0 -10000.0)
     (+ z x)
     (if (<= t_0 -5e-5)
       (sin y)
       (if (<= t_0 2e-7) (+ (+ z y) x) (if (<= t_0 1.0) (sin y) (+ z x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (z * cos(y));
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = z + x;
	} else if (t_0 <= -5e-5) {
		tmp = sin(y);
	} else if (t_0 <= 2e-7) {
		tmp = (z + y) + x;
	} else if (t_0 <= 1.0) {
		tmp = sin(y);
	} else {
		tmp = z + 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) :: tmp
    t_0 = (x + sin(y)) + (z * cos(y))
    if (t_0 <= (-10000.0d0)) then
        tmp = z + x
    else if (t_0 <= (-5d-5)) then
        tmp = sin(y)
    else if (t_0 <= 2d-7) then
        tmp = (z + y) + x
    else if (t_0 <= 1.0d0) then
        tmp = sin(y)
    else
        tmp = z + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + Math.sin(y)) + (z * Math.cos(y));
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = z + x;
	} else if (t_0 <= -5e-5) {
		tmp = Math.sin(y);
	} else if (t_0 <= 2e-7) {
		tmp = (z + y) + x;
	} else if (t_0 <= 1.0) {
		tmp = Math.sin(y);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.sin(y)) + (z * math.cos(y))
	tmp = 0
	if t_0 <= -10000.0:
		tmp = z + x
	elif t_0 <= -5e-5:
		tmp = math.sin(y)
	elif t_0 <= 2e-7:
		tmp = (z + y) + x
	elif t_0 <= 1.0:
		tmp = math.sin(y)
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	tmp = 0.0
	if (t_0 <= -10000.0)
		tmp = Float64(z + x);
	elseif (t_0 <= -5e-5)
		tmp = sin(y);
	elseif (t_0 <= 2e-7)
		tmp = Float64(Float64(z + y) + x);
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + sin(y)) + (z * cos(y));
	tmp = 0.0;
	if (t_0 <= -10000.0)
		tmp = z + x;
	elseif (t_0 <= -5e-5)
		tmp = sin(y);
	elseif (t_0 <= 2e-7)
		tmp = (z + y) + x;
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10000.0], N[(z + x), $MachinePrecision], If[LessEqual[t$95$0, -5e-5], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 2e-7], N[(N[(z + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[y], $MachinePrecision], N[(z + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -1e4 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 77.4

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

   4. Applied rewrites 77.4%

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

   if -1e4 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -5.00000000000000024e-5 or 1.9999999999999999e-7 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto z \cdot \cos y + \color{blue}{\sin y} \]

      2. *-commutative N/A

         \[\leadsto \cos y \cdot z + \sin \color{blue}{y} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, \color{blue}{z}, \sin y\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \sin y\right) \]

      5. lift-sin.f64 94.9

         \[\leadsto \mathsf{fma}\left(\cos y, z, \sin y\right) \]

   4. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y\right)} \]

   5. Taylor expanded in z around 0

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

      1. lift-sin.f64 89.9

         \[\leadsto \sin y \]

   7. Applied rewrites 89.9%

      \[\leadsto \sin y \]

   if -5.00000000000000024e-5 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1.9999999999999999e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 6: 81.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y + x\\ \mathbf{if}\;y \leq -0.005:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.056:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, z + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (sin y) x)))
   (if (<= y -0.005)
     t_0
     (if (<= y 0.056) (fma (fma (* z y) -0.5 1.0) y (+ z x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) + x;
	double tmp;
	if (y <= -0.005) {
		tmp = t_0;
	} else if (y <= 0.056) {
		tmp = fma(fma((z * y), -0.5, 1.0), y, (z + x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) + x)
	tmp = 0.0
	if (y <= -0.005)
		tmp = t_0;
	elseif (y <= 0.056)
		tmp = fma(fma(Float64(z * y), -0.5, 1.0), y, Float64(z + x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -0.005], t$95$0, If[LessEqual[y, 0.056], N[(N[(N[(z * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0050000000000000001 or 0.0560000000000000012 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sin.f64 62.3

         \[\leadsto \sin y + x \]

   4. Applied rewrites 62.3%

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

   if -0.0050000000000000001 < y < 0.0560000000000000012

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 7: 71.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -37000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 3500:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -37000.0)
   (+ z x)
   (if (<= y 3500.0)
     (+ (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y z) x)
     (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -37000.0) {
		tmp = z + x;
	} else if (y <= 3500.0) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, z) + x;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -37000.0)
		tmp = Float64(z + x);
	elseif (y <= 3500.0)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, z) + x);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -37000.0], N[(z + x), $MachinePrecision], If[LessEqual[y, 3500.0], N[(N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + z), $MachinePrecision] + x), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -37000 or 3500 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 42.6

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

   4. Applied rewrites 42.6%

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

   if -37000 < y < 3500

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

Alternative 8: 71.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -250:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 1850:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, z + x\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -250.0)
   (+ z x)
   (if (<= y 1850.0) (fma (fma (* z y) -0.5 1.0) y (+ z x)) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -250.0) {
		tmp = z + x;
	} else if (y <= 1850.0) {
		tmp = fma(fma((z * y), -0.5, 1.0), y, (z + x));
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -250.0)
		tmp = Float64(z + x);
	elseif (y <= 1850.0)
		tmp = fma(fma(Float64(z * y), -0.5, 1.0), y, Float64(z + x));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -250.0], N[(z + x), $MachinePrecision], If[LessEqual[y, 1850.0], N[(N[(N[(z * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -250 or 1850 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 42.6

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

   4. Applied rewrites 42.6%

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

   if -250 < y < 1850

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

Alternative 9: 71.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;z + x\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-32}:\\ \;\;\;\;\left(z + y\right) + x\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (<= t_0 -0.02) (+ z x) (if (<= t_0 4e-32) (+ (+ z y) x) (+ z x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (z * cos(y));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = z + x;
	} else if (t_0 <= 4e-32) {
		tmp = (z + y) + x;
	} else {
		tmp = z + 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) :: tmp
    t_0 = (x + sin(y)) + (z * cos(y))
    if (t_0 <= (-0.02d0)) then
        tmp = z + x
    else if (t_0 <= 4d-32) then
        tmp = (z + y) + x
    else
        tmp = z + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + Math.sin(y)) + (z * Math.cos(y));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = z + x;
	} else if (t_0 <= 4e-32) {
		tmp = (z + y) + x;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.sin(y)) + (z * math.cos(y))
	tmp = 0
	if t_0 <= -0.02:
		tmp = z + x
	elif t_0 <= 4e-32:
		tmp = (z + y) + x
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(z + x);
	elseif (t_0 <= 4e-32)
		tmp = Float64(Float64(z + y) + x);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + sin(y)) + (z * cos(y));
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = z + x;
	elseif (t_0 <= 4e-32)
		tmp = (z + y) + x;
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(z + x), $MachinePrecision], If[LessEqual[t$95$0, 4e-32], N[(N[(z + y), $MachinePrecision] + x), $MachinePrecision], N[(z + x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.0200000000000000004 or 4.00000000000000022e-32 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 67.7

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

   4. Applied rewrites 67.7%

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

   if -0.0200000000000000004 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 4.00000000000000022e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 10: 67.5% accurate, 19.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 67.5

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

4. Applied rewrites 67.5%

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

Alternative 11: 59.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-21}:\\ \;\;\;\;z + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-17) x (if (<= x 1.65e-21) (+ z y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-17) {
		tmp = x;
	} else if (x <= 1.65e-21) {
		tmp = z + y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.5d-17)) then
        tmp = x
    else if (x <= 1.65d-21) then
        tmp = z + y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-17) {
		tmp = x;
	} else if (x <= 1.65e-21) {
		tmp = z + y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5e-17:
		tmp = x
	elif x <= 1.65e-21:
		tmp = z + y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e-17)
		tmp = x;
	elseif (x <= 1.65e-21)
		tmp = Float64(z + y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5e-17)
		tmp = x;
	elseif (x <= 1.65e-21)
		tmp = z + y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e-17], x, If[LessEqual[x, 1.65e-21], N[(z + y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000003e-17 or 1.65000000000000004e-21 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 72.2%

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

   if -1.50000000000000003e-17 < x < 1.65000000000000004e-21

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto z \cdot \cos y + \color{blue}{\sin y} \]

      2. *-commutative N/A

         \[\leadsto \cos y \cdot z + \sin \color{blue}{y} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, \color{blue}{z}, \sin y\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \sin y\right) \]

      5. lift-sin.f64 92.5

         \[\leadsto \mathsf{fma}\left(\cos y, z, \sin y\right) \]

   4. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y\right)} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto z + y \]

      2. lower-+.f64 45.5

         \[\leadsto z + y \]

   7. Applied rewrites 45.5%

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

Alternative 12: 56.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-17}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-17) x (if (<= x 1.35e-17) z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-17) {
		tmp = x;
	} else if (x <= 1.35e-17) {
		tmp = z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.5d-17)) then
        tmp = x
    else if (x <= 1.35d-17) then
        tmp = z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-17) {
		tmp = x;
	} else if (x <= 1.35e-17) {
		tmp = z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5e-17:
		tmp = x
	elif x <= 1.35e-17:
		tmp = z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e-17)
		tmp = x;
	elseif (x <= 1.35e-17)
		tmp = z;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5e-17)
		tmp = x;
	elseif (x <= 1.35e-17)
		tmp = z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e-17], x, If[LessEqual[x, 1.35e-17], z, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000003e-17 or 1.3500000000000001e-17 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 72.6%

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

   if -1.50000000000000003e-17 < x < 1.3500000000000001e-17

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto z \cdot \cos y + \color{blue}{\sin y} \]

      2. *-commutative N/A

         \[\leadsto \cos y \cdot z + \sin \color{blue}{y} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, \color{blue}{z}, \sin y\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \sin y\right) \]

      5. lift-sin.f64 92.4

         \[\leadsto \mathsf{fma}\left(\cos y, z, \sin y\right) \]

   4. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto z \]
   6. Step-by-step derivation

   7. Applied rewrites 38.3%

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

Alternative 13: 43.1% accurate, 73.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 43.1%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C"
  :precision binary64
  (+ (+ x (sin y)) (* z (cos y))))