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

Percentage Accurate: 99.9% → 99.9%

Time: 4.0s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - z \cdot \sin y\\ t_1 := x - \sin y \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 20000000:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* z (sin y)))) (t_1 (- x (* (sin y) z))))
   (if (<= t_0 -2e+14) t_1 (if (<= t_0 20000000.0) (+ (cos y) x) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (z * sin(y));
	double t_1 = x - (sin(y) * z);
	double tmp;
	if (t_0 <= -2e+14) {
		tmp = t_1;
	} else if (t_0 <= 20000000.0) {
		tmp = cos(y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + cos(y)) - (z * sin(y))
    t_1 = x - (sin(y) * z)
    if (t_0 <= (-2d+14)) then
        tmp = t_1
    else if (t_0 <= 20000000.0d0) then
        tmp = cos(y) + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + Math.cos(y)) - (z * Math.sin(y));
	double t_1 = x - (Math.sin(y) * z);
	double tmp;
	if (t_0 <= -2e+14) {
		tmp = t_1;
	} else if (t_0 <= 20000000.0) {
		tmp = Math.cos(y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.cos(y)) - (z * math.sin(y))
	t_1 = x - (math.sin(y) * z)
	tmp = 0
	if t_0 <= -2e+14:
		tmp = t_1
	elif t_0 <= 20000000.0:
		tmp = math.cos(y) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
	t_1 = Float64(x - Float64(sin(y) * z))
	tmp = 0.0
	if (t_0 <= -2e+14)
		tmp = t_1;
	elseif (t_0 <= 20000000.0)
		tmp = Float64(cos(y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + cos(y)) - (z * sin(y));
	t_1 = x - (sin(y) * z);
	tmp = 0.0;
	if (t_0 <= -2e+14)
		tmp = t_1;
	elseif (t_0 <= 20000000.0)
		tmp = cos(y) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+14], t$95$1, If[LessEqual[t$95$0, 20000000.0], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -2e14 or 2e7 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if -2e14 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 2e7

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 96.2

         \[\leadsto \cos y + x \]

   4. Applied rewrites 96.2%

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

Alternative 3: 81.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.48 \cdot 10^{+184}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{+59}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.48e+184)
   (* (- z) (sin y))
   (if (<= z -5.6e+59) (- x (fma z y -1.0)) (+ (cos y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.48e+184) {
		tmp = -z * sin(y);
	} else if (z <= -5.6e+59) {
		tmp = x - fma(z, y, -1.0);
	} else {
		tmp = cos(y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.48e+184)
		tmp = Float64(Float64(-z) * sin(y));
	elseif (z <= -5.6e+59)
		tmp = Float64(x - fma(z, y, -1.0));
	else
		tmp = Float64(cos(y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.48e+184], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -5.6e+59], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.48e184

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-sin.f64 72.2

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

   4. Applied rewrites 72.2%

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

   if -1.48e184 < z < -5.5999999999999996e59

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.2

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

   4. Applied rewrites 52.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. negate-sub N/A

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

      8. lower--.f64 N/A

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

      9. negate-sub N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 52.2

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

   6. Applied rewrites 52.2%

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

   if -5.5999999999999996e59 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 81.1

         \[\leadsto \cos y + x \]

   4. Applied rewrites 81.1%

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

Alternative 4: 77.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y + x\\ \mathbf{if}\;y \leq -0.032:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1150000000:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot y\right) \cdot 0.16666666666666666\right) \cdot y - z, y, x\right) - -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (cos y) x)))
   (if (<= y -0.032)
     t_0
     (if (<= y 1150000000.0)
       (- (fma (- (* (* (* z y) 0.16666666666666666) y) z) y x) -1.0)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) + x;
	double tmp;
	if (y <= -0.032) {
		tmp = t_0;
	} else if (y <= 1150000000.0) {
		tmp = fma(((((z * y) * 0.16666666666666666) * y) - z), y, x) - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) + x)
	tmp = 0.0
	if (y <= -0.032)
		tmp = t_0;
	elseif (y <= 1150000000.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(z * y) * 0.16666666666666666) * y) - z), y, x) - -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -0.032], t$95$0, If[LessEqual[y, 1150000000.0], N[(N[(N[(N[(N[(N[(z * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision] - -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.032000000000000001 or 1.15e9 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 63.0

         \[\leadsto \cos y + x \]

   4. Applied rewrites 63.0%

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

   if -0.032000000000000001 < y < 1.15e9

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 98.8

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

   7. Applied rewrites 98.8%

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

Alternative 5: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \mathsf{fma}\left(z, y, -1\right)\\ t_1 := \left(x + \cos y\right) - z \cdot \sin y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq -400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.999:\\ \;\;\;\;\cos y\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+205}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (fma z y -1.0))) (t_1 (- (+ x (cos y)) (* z (sin y)))))
   (if (<= t_1 -5e+77)
     x
     (if (<= t_1 -400.0)
       t_0
       (if (<= t_1 0.999) (cos y) (if (<= t_1 5e+205) t_0 (- x -1.0)))))))

C

double code(double x, double y, double z) {
	double t_0 = x - fma(z, y, -1.0);
	double t_1 = (x + cos(y)) - (z * sin(y));
	double tmp;
	if (t_1 <= -5e+77) {
		tmp = x;
	} else if (t_1 <= -400.0) {
		tmp = t_0;
	} else if (t_1 <= 0.999) {
		tmp = cos(y);
	} else if (t_1 <= 5e+205) {
		tmp = t_0;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x - fma(z, y, -1.0))
	t_1 = Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
	tmp = 0.0
	if (t_1 <= -5e+77)
		tmp = x;
	elseif (t_1 <= -400.0)
		tmp = t_0;
	elseif (t_1 <= 0.999)
		tmp = cos(y);
	elseif (t_1 <= 5e+205)
		tmp = t_0;
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+77], x, If[LessEqual[t$95$1, -400.0], t$95$0, If[LessEqual[t$95$1, 0.999], N[Cos[y], $MachinePrecision], If[LessEqual[t$95$1, 5e+205], t$95$0, N[(x - -1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -5.00000000000000004e77

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 61.5%

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

   if -5.00000000000000004e77 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -400 or 0.998999999999999999 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 5.0000000000000002e205

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. negate-sub N/A

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

      8. lower--.f64 N/A

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

      9. negate-sub N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 78.2

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

   6. Applied rewrites 78.2%

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

   if -400 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.998999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-cos.f64 96.9

         \[\leadsto \cos y + x \]

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in x around 0

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

      1. lift-cos.f64 93.2

         \[\leadsto \cos y \]

   7. Applied rewrites 93.2%

      \[\leadsto \cos y \]

   if 5.0000000000000002e205 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 66.3

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

   4. Applied rewrites 66.3%

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

Alternative 6: 69.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+43}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5\right) \cdot y - z, y, x\right) - -1\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.8e+43)
   (- x -1.0)
   (if (<= y 1.12e+30)
     (- (fma (- (* (fma 0.16666666666666666 (* z y) -0.5) y) z) y x) -1.0)
     (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.8e+43) {
		tmp = x - -1.0;
	} else if (y <= 1.12e+30) {
		tmp = fma(((fma(0.16666666666666666, (z * y), -0.5) * y) - z), y, x) - -1.0;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.8e+43)
		tmp = Float64(x - -1.0);
	elseif (y <= 1.12e+30)
		tmp = Float64(fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), -0.5) * y) - z), y, x) - -1.0);
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.8e+43], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 1.12e+30], N[(N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + -0.5), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision] - -1.0), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000019e43 or 1.12e30 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 40.1

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

   4. Applied rewrites 40.1%

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

   if -2.80000000000000019e43 < y < 1.12e30

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 92.3%

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

Alternative 7: 69.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+42}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot y\right) \cdot 0.16666666666666666\right) \cdot y - z, y, x\right) - -1\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.9e+42)
   (- x -1.0)
   (if (<= y 1.12e+30)
     (- (fma (- (* (* (* z y) 0.16666666666666666) y) z) y x) -1.0)
     (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.9e+42) {
		tmp = x - -1.0;
	} else if (y <= 1.12e+30) {
		tmp = fma(((((z * y) * 0.16666666666666666) * y) - z), y, x) - -1.0;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.9e+42)
		tmp = Float64(x - -1.0);
	elseif (y <= 1.12e+30)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(z * y) * 0.16666666666666666) * y) - z), y, x) - -1.0);
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.9e+42], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 1.12e+30], N[(N[(N[(N[(N[(N[(z * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision] - -1.0), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8999999999999999e42 or 1.12e30 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 40.1

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

   4. Applied rewrites 40.1%

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

   if -1.8999999999999999e42 < y < 1.12e30

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 92.8

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

   7. Applied rewrites 92.8%

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

Alternative 8: 69.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+43}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, x\right) - -1\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e+43)
   (- x -1.0)
   (if (<= y 1.8e+24) (- (fma (- (* -0.5 y) z) y x) -1.0) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+43) {
		tmp = x - -1.0;
	} else if (y <= 1.8e+24) {
		tmp = fma(((-0.5 * y) - z), y, x) - -1.0;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e+43)
		tmp = Float64(x - -1.0);
	elseif (y <= 1.8e+24)
		tmp = Float64(fma(Float64(Float64(-0.5 * y) - z), y, x) - -1.0);
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.1e+43], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 1.8e+24], N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - z), $MachinePrecision] * y + x), $MachinePrecision] - -1.0), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1000000000000002e43 or 1.79999999999999992e24 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 39.9

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

   4. Applied rewrites 39.9%

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

   if -3.1000000000000002e43 < y < 1.79999999999999992e24

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 92.8

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

   4. Applied rewrites 92.8%

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

Alternative 9: 69.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4400000000000:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+72}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4400000000000.0)
   (- x -1.0)
   (if (<= y 1.2e+72) (- x (fma z y -1.0)) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4400000000000.0) {
		tmp = x - -1.0;
	} else if (y <= 1.2e+72) {
		tmp = x - fma(z, y, -1.0);
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4400000000000.0)
		tmp = Float64(x - -1.0);
	elseif (y <= 1.2e+72)
		tmp = Float64(x - fma(z, y, -1.0));
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4400000000000.0], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 1.2e+72], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4e12 or 1.20000000000000005e72 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 40.0

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

   4. Applied rewrites 40.0%

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

   if -4.4e12 < y < 1.20000000000000005e72

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. negate-sub N/A

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

      8. lower--.f64 N/A

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

      9. negate-sub N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 91.8

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

   6. Applied rewrites 91.8%

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

Alternative 10: 66.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-16}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;1 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.02e-16)
   (- x -1.0)
   (if (<= x 2.2e-9) (- 1.0 (* z y)) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.02e-16) {
		tmp = x - -1.0;
	} else if (x <= 2.2e-9) {
		tmp = 1.0 - (z * y);
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.02d-16)) then
        tmp = x - (-1.0d0)
    else if (x <= 2.2d-9) then
        tmp = 1.0d0 - (z * y)
    else
        tmp = x - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.02e-16) {
		tmp = x - -1.0;
	} else if (x <= 2.2e-9) {
		tmp = 1.0 - (z * y);
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.02e-16:
		tmp = x - -1.0
	elif x <= 2.2e-9:
		tmp = 1.0 - (z * y)
	else:
		tmp = x - -1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.02e-16)
		tmp = Float64(x - -1.0);
	elseif (x <= 2.2e-9)
		tmp = Float64(1.0 - Float64(z * y));
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.02e-16)
		tmp = x - -1.0;
	elseif (x <= 2.2e-9)
		tmp = 1.0 - (z * y);
	else
		tmp = x - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.02e-16], N[(x - -1.0), $MachinePrecision], If[LessEqual[x, 2.2e-9], N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0200000000000001e-16 or 2.1999999999999998e-9 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 81.3

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

   4. Applied rewrites 81.3%

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

   if -1.0200000000000001e-16 < x < 2.1999999999999998e-9

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. negate-sub-reverse N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 51.4

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

   7. Applied rewrites 51.4%

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

Alternative 11: 62.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+217}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.02e+217) (* (- z) y) (- x -1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.02e+217) {
		tmp = -z * y;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.02d+217)) then
        tmp = -z * y
    else
        tmp = x - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.02e+217) {
		tmp = -z * y;
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.02e+217:
		tmp = -z * y
	else:
		tmp = x - -1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.02e+217)
		tmp = Float64(Float64(-z) * y);
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.02e+217)
		tmp = -z * y;
	else
		tmp = x - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.02e+217], N[((-z) * y), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02e217

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-sin.f64 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 32.4%

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

   if -1.02e217 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 64.6

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

   4. Applied rewrites 64.6%

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

Alternative 12: 61.4% accurate, 20.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. negate-sub-reverse N/A

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

   4. lower--.f64 61.4

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

4. Applied rewrites 61.4%

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

Alternative 13: 42.2% accurate, 72.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 42.2%

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

Reproduce

herbie shell --seed 2025110 
(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))))