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

Percentage Accurate: 99.9% → 99.9%

Time: 9.7s

Alternatives: 17

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

real(8) function code(x, y, z)
    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

real(8) function code(x, y, z)
    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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]
2. Step-by-step derivation

   1. cancel-sign-sub-inv 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+l+ 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-lft-neg-out 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. sin-neg 99.9%

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

   8. fma-define 99.9%

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

   9. sin-neg 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\cos y + \mathsf{fma}\left(z, -\sin y, x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;x \leq -9 \cdot 10^{-7} \lor \neg \left(x \leq 3 \cdot 10^{-11}\right):\\ \;\;\;\;\left(x + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (or (<= x -9e-7) (not (<= x 3e-11)))
     (- (+ x 1.0) t_0)
     (- (cos y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if ((x <= -9e-7) || !(x <= 3e-11)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = cos(y) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * sin(y)
    if ((x <= (-9d-7)) .or. (.not. (x <= 3d-11))) then
        tmp = (x + 1.0d0) - t_0
    else
        tmp = cos(y) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double tmp;
	if ((x <= -9e-7) || !(x <= 3e-11)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = Math.cos(y) - t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	tmp = 0
	if (x <= -9e-7) or not (x <= 3e-11):
		tmp = (x + 1.0) - t_0
	else:
		tmp = math.cos(y) - t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if ((x <= -9e-7) || !(x <= 3e-11))
		tmp = Float64(Float64(x + 1.0) - t_0);
	else
		tmp = Float64(cos(y) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	tmp = 0.0;
	if ((x <= -9e-7) || ~((x <= 3e-11)))
		tmp = (x + 1.0) - t_0;
	else
		tmp = cos(y) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -9e-7], N[Not[LessEqual[x, 3e-11]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.99999999999999959e-7 or 3e-11 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.5%

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

   if -8.99999999999999959e-7 < x < 3e-11

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. sub-neg 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-7} \lor \neg \left(x \leq 3 \cdot 10^{-11}\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;x \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(z, -\sin y, x\right) + 1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\cos y - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= x -9e-7)
     (+ (fma z (- (sin y)) x) 1.0)
     (if (<= x 2.8e-9) (- (cos y) t_0) (- (+ x 1.0) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (x <= -9e-7) {
		tmp = fma(z, -sin(y), x) + 1.0;
	} else if (x <= 2.8e-9) {
		tmp = cos(y) - t_0;
	} else {
		tmp = (x + 1.0) - t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (x <= -9e-7)
		tmp = Float64(fma(z, Float64(-sin(y)), x) + 1.0);
	elseif (x <= 2.8e-9)
		tmp = Float64(cos(y) - t_0);
	else
		tmp = Float64(Float64(x + 1.0) - t_0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e-7], N[(N[(z * (-N[Sin[y], $MachinePrecision]) + x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 2.8e-9], N[(N[Cos[y], $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.99999999999999959e-7

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.1%

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

   if -8.99999999999999959e-7 < x < 2.79999999999999984e-9

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. sub-neg 99.3%

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

   7. Simplified 99.3%

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

   if 2.79999999999999984e-9 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto \left(x + \color{blue}{1}\right) - z \cdot \sin y \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(z, -\sin y, x\right) + 1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 80.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+65}:\\ \;\;\;\;1 - z \cdot \sin y\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+231}:\\ \;\;\;\;\cos y + x\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+258}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.6e+65)
   (- 1.0 (* z (sin y)))
   (if (<= z 4.9e+231)
     (+ (cos y) x)
     (if (<= z 1.5e+258)
       (* z (- (sin y)))
       (+
        1.0
        (+ x (* y (- (* y (- (* 0.16666666666666666 (* y z)) 0.5)) z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.6e+65) {
		tmp = 1.0 - (z * sin(y));
	} else if (z <= 4.9e+231) {
		tmp = cos(y) + x;
	} else if (z <= 1.5e+258) {
		tmp = z * -sin(y);
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.6d+65)) then
        tmp = 1.0d0 - (z * sin(y))
    else if (z <= 4.9d+231) then
        tmp = cos(y) + x
    else if (z <= 1.5d+258) then
        tmp = z * -sin(y)
    else
        tmp = 1.0d0 + (x + (y * ((y * ((0.16666666666666666d0 * (y * z)) - 0.5d0)) - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.6e+65) {
		tmp = 1.0 - (z * Math.sin(y));
	} else if (z <= 4.9e+231) {
		tmp = Math.cos(y) + x;
	} else if (z <= 1.5e+258) {
		tmp = z * -Math.sin(y);
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.6e+65:
		tmp = 1.0 - (z * math.sin(y))
	elif z <= 4.9e+231:
		tmp = math.cos(y) + x
	elif z <= 1.5e+258:
		tmp = z * -math.sin(y)
	else:
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.6e+65)
		tmp = Float64(1.0 - Float64(z * sin(y)));
	elseif (z <= 4.9e+231)
		tmp = Float64(cos(y) + x);
	elseif (z <= 1.5e+258)
		tmp = Float64(z * Float64(-sin(y)));
	else
		tmp = Float64(1.0 + Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(0.16666666666666666 * Float64(y * z)) - 0.5)) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.6e+65)
		tmp = 1.0 - (z * sin(y));
	elseif (z <= 4.9e+231)
		tmp = cos(y) + x;
	elseif (z <= 1.5e+258)
		tmp = z * -sin(y);
	else
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.6e+65], N[(1.0 - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.9e+231], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.5e+258], N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision], N[(1.0 + N[(x + N[(y * N[(N[(y * N[(N[(0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.60000000000000003e65

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. sin-neg 99.8%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 72.4%

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

      1. neg-mul-1 72.4%

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

      2. sub-neg 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in y around 0 72.4%

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

   if -2.60000000000000003e65 < z < 4.90000000000000021e231

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 90.9%

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

   if 4.90000000000000021e231 < z < 1.5e258

   1. Initial program 99.6%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft-neg-out 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. sin-neg 99.6%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 93.8%

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

      1. neg-mul-1 93.8%

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

      2. distribute-rgt-neg-in 93.8%

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

   7. Simplified 93.8%

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

   if 1.5e258 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 93.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+65}:\\ \;\;\;\;1 - z \cdot \sin y\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+231}:\\ \;\;\;\;\cos y + x\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+258}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-\sin y\right)\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+231}:\\ \;\;\;\;\cos y + x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+258}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (sin y)))))
   (if (<= z -1.05e+113)
     t_0
     (if (<= z 4.9e+231)
       (+ (cos y) x)
       (if (<= z 1.45e+258)
         t_0
         (+
          1.0
          (+ x (* y (- (* y (- (* 0.16666666666666666 (* y z)) 0.5)) z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -sin(y);
	double tmp;
	if (z <= -1.05e+113) {
		tmp = t_0;
	} else if (z <= 4.9e+231) {
		tmp = cos(y) + x;
	} else if (z <= 1.45e+258) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * -sin(y)
    if (z <= (-1.05d+113)) then
        tmp = t_0
    else if (z <= 4.9d+231) then
        tmp = cos(y) + x
    else if (z <= 1.45d+258) then
        tmp = t_0
    else
        tmp = 1.0d0 + (x + (y * ((y * ((0.16666666666666666d0 * (y * z)) - 0.5d0)) - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * -Math.sin(y);
	double tmp;
	if (z <= -1.05e+113) {
		tmp = t_0;
	} else if (z <= 4.9e+231) {
		tmp = Math.cos(y) + x;
	} else if (z <= 1.45e+258) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -math.sin(y)
	tmp = 0
	if z <= -1.05e+113:
		tmp = t_0
	elif z <= 4.9e+231:
		tmp = math.cos(y) + x
	elif z <= 1.45e+258:
		tmp = t_0
	else:
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-sin(y)))
	tmp = 0.0
	if (z <= -1.05e+113)
		tmp = t_0;
	elseif (z <= 4.9e+231)
		tmp = Float64(cos(y) + x);
	elseif (z <= 1.45e+258)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(0.16666666666666666 * Float64(y * z)) - 0.5)) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -sin(y);
	tmp = 0.0;
	if (z <= -1.05e+113)
		tmp = t_0;
	elseif (z <= 4.9e+231)
		tmp = cos(y) + x;
	elseif (z <= 1.45e+258)
		tmp = t_0;
	else
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[z, -1.05e+113], t$95$0, If[LessEqual[z, 4.9e+231], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.45e+258], t$95$0, N[(1.0 + N[(x + N[(y * N[(N[(y * N[(N[(0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0499999999999999e113 or 4.90000000000000021e231 < z < 1.4500000000000001e258

   1. Initial program 99.7%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-lft-neg-out 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. sin-neg 99.7%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 71.8%

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

      1. neg-mul-1 71.8%

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

      2. distribute-rgt-neg-in 71.8%

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

   7. Simplified 71.8%

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

   if -1.0499999999999999e113 < z < 4.90000000000000021e231

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 89.0%

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

   if 1.4500000000000001e258 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 93.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+113}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+231}:\\ \;\;\;\;\cos y + x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+258}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.226 \lor \neg \left(z \leq 7 \cdot 10^{-12}\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.226) (not (<= z 7e-12)))
   (- (+ x 1.0) (* z (sin y)))
   (+ (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.226) || !(z <= 7e-12)) {
		tmp = (x + 1.0) - (z * sin(y));
	} else {
		tmp = cos(y) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-0.226d0)) .or. (.not. (z <= 7d-12))) then
        tmp = (x + 1.0d0) - (z * sin(y))
    else
        tmp = cos(y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.226) || !(z <= 7e-12)) {
		tmp = (x + 1.0) - (z * Math.sin(y));
	} else {
		tmp = Math.cos(y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -0.226) or not (z <= 7e-12):
		tmp = (x + 1.0) - (z * math.sin(y))
	else:
		tmp = math.cos(y) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.226) || !(z <= 7e-12))
		tmp = Float64(Float64(x + 1.0) - Float64(z * sin(y)));
	else
		tmp = Float64(cos(y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -0.226) || ~((z <= 7e-12)))
		tmp = (x + 1.0) - (z * sin(y));
	else
		tmp = cos(y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.226], N[Not[LessEqual[z, 7e-12]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.226000000000000006 or 7.0000000000000001e-12 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 99.2%

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

   if -0.226000000000000006 < z < 7.0000000000000001e-12

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-neg-out 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. sin-neg 100.0%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.226 \lor \neg \left(z \leq 7 \cdot 10^{-12}\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+64} \lor \neg \left(z \leq 2.6 \cdot 10^{+29}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.3e+64) (not (<= z 2.6e+29)))
   (- x (* z (sin y)))
   (+ (cos y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3e+64) || !(z <= 2.6e+29)) {
		tmp = x - (z * sin(y));
	} else {
		tmp = cos(y) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.3d+64)) .or. (.not. (z <= 2.6d+29))) then
        tmp = x - (z * sin(y))
    else
        tmp = cos(y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3e+64) || !(z <= 2.6e+29)) {
		tmp = x - (z * Math.sin(y));
	} else {
		tmp = Math.cos(y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.3e+64) or not (z <= 2.6e+29):
		tmp = x - (z * math.sin(y))
	else:
		tmp = math.cos(y) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.3e+64) || !(z <= 2.6e+29))
		tmp = Float64(x - Float64(z * sin(y)));
	else
		tmp = Float64(cos(y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.3e+64) || ~((z <= 2.6e+29)))
		tmp = x - (z * sin(y));
	else
		tmp = cos(y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.3e+64], N[Not[LessEqual[z, 2.6e+29]], $MachinePrecision]], N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999998e64 or 2.6e29 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 99.8%

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

   4. Taylor expanded in x around inf 91.4%

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

   if -1.29999999999999998e64 < z < 2.6e29

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-neg-out 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. sin-neg 100.0%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 96.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+64} \lor \neg \left(z \leq 2.6 \cdot 10^{+29}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -106000 \lor \neg \left(y \leq 4.4 \cdot 10^{-8}\right):\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -106000.0) (not (<= y 4.4e-8)))
   (+ (cos y) x)
   (+ 1.0 (+ x (* y (- (* y (- (* 0.16666666666666666 (* y z)) 0.5)) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -106000.0) || !(y <= 4.4e-8)) {
		tmp = cos(y) + x;
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-106000.0d0)) .or. (.not. (y <= 4.4d-8))) then
        tmp = cos(y) + x
    else
        tmp = 1.0d0 + (x + (y * ((y * ((0.16666666666666666d0 * (y * z)) - 0.5d0)) - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -106000.0) || !(y <= 4.4e-8)) {
		tmp = Math.cos(y) + x;
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -106000.0) or not (y <= 4.4e-8):
		tmp = math.cos(y) + x
	else:
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -106000.0) || !(y <= 4.4e-8))
		tmp = Float64(cos(y) + x);
	else
		tmp = Float64(1.0 + Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(0.16666666666666666 * Float64(y * z)) - 0.5)) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -106000.0) || ~((y <= 4.4e-8)))
		tmp = cos(y) + x;
	else
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -106000.0], N[Not[LessEqual[y, 4.4e-8]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], N[(1.0 + N[(x + N[(y * N[(N[(y * N[(N[(0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -106000 or 4.3999999999999997e-8 < y

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. sin-neg 99.8%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 64.6%

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

   if -106000 < y < 4.3999999999999997e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -106000 \lor \neg \left(y \leq 4.4 \cdot 10^{-8}\right):\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-7}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-11}:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9e-7) (+ x 1.0) (if (<= x 3e-11) (cos y) (* x (+ 1.0 (/ 1.0 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e-7) {
		tmp = x + 1.0;
	} else if (x <= 3e-11) {
		tmp = cos(y);
	} else {
		tmp = x * (1.0 + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-9d-7)) then
        tmp = x + 1.0d0
    else if (x <= 3d-11) then
        tmp = cos(y)
    else
        tmp = x * (1.0d0 + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e-7) {
		tmp = x + 1.0;
	} else if (x <= 3e-11) {
		tmp = Math.cos(y);
	} else {
		tmp = x * (1.0 + (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9e-7:
		tmp = x + 1.0
	elif x <= 3e-11:
		tmp = math.cos(y)
	else:
		tmp = x * (1.0 + (1.0 / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9e-7)
		tmp = Float64(x + 1.0);
	elseif (x <= 3e-11)
		tmp = cos(y);
	else
		tmp = Float64(x * Float64(1.0 + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9e-7)
		tmp = x + 1.0;
	elseif (x <= 3e-11)
		tmp = cos(y);
	else
		tmp = x * (1.0 + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9e-7], N[(x + 1.0), $MachinePrecision], If[LessEqual[x, 3e-11], N[Cos[y], $MachinePrecision], N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.99999999999999959e-7

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 81.5%

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

   if -8.99999999999999959e-7 < x < 3e-11

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. sub-neg 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in z around 0 68.5%

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

   if 3e-11 < x

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-neg-out 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. sin-neg 100.0%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-undefine 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-+l+ 100.0%

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

      5. distribute-lft-neg-in 100.0%

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

      6. add-cube-cbrt 99.5%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}\right)} + \left(x + \cos y\right) \]

      7. associate-*r* 99.5%

         \[\leadsto \color{blue}{\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)\right) \cdot \sqrt[3]{\sin y}} + \left(x + \cos y\right) \]

      8. fma-define 99.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right), \sqrt[3]{\sin y}, x + \cos y\right)} \]

      9. pow2 99.5%

         \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot \color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y}, x + \cos y\right) \]

      10. +-commutative 99.5%

          \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, \color{blue}{\cos y + x}\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, \cos y + x\right)} \]

   7. Taylor expanded in y around 0 82.8%

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

      1. +-commutative 82.8%

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

   9. Simplified 82.8%

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

   10. Taylor expanded in x around inf 82.8%

       \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-7}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-11}:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4200000000 \lor \neg \left(y \leq 10.8\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot -0.5 - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4200000000.0) (not (<= y 10.8)))
   (+ x 1.0)
   (+ 1.0 (+ x (* y (- (* y -0.5) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4200000000.0) || !(y <= 10.8)) {
		tmp = x + 1.0;
	} else {
		tmp = 1.0 + (x + (y * ((y * -0.5) - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-4200000000.0d0)) .or. (.not. (y <= 10.8d0))) then
        tmp = x + 1.0d0
    else
        tmp = 1.0d0 + (x + (y * ((y * (-0.5d0)) - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4200000000.0) || !(y <= 10.8)) {
		tmp = x + 1.0;
	} else {
		tmp = 1.0 + (x + (y * ((y * -0.5) - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4200000000.0) or not (y <= 10.8):
		tmp = x + 1.0
	else:
		tmp = 1.0 + (x + (y * ((y * -0.5) - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4200000000.0) || !(y <= 10.8))
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(1.0 + Float64(x + Float64(y * Float64(Float64(y * -0.5) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4200000000.0) || ~((y <= 10.8)))
		tmp = x + 1.0;
	else
		tmp = 1.0 + (x + (y * ((y * -0.5) - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4200000000.0], N[Not[LessEqual[y, 10.8]], $MachinePrecision]], N[(x + 1.0), $MachinePrecision], N[(1.0 + N[(x + N[(y * N[(N[(y * -0.5), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e9 or 10.800000000000001 < y

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. sin-neg 99.8%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 36.6%

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

   if -4.2e9 < y < 10.800000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.3%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4200000000 \lor \neg \left(y \leq 10.8\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot -0.5 - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{+23}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+81}:\\ \;\;\;\;1 + \left(x - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -9.4e+23)
   (+ x 1.0)
   (if (<= y 5.2e+81) (+ 1.0 (- x (* y z))) (* x (+ 1.0 (/ 1.0 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -9.4e+23) {
		tmp = x + 1.0;
	} else if (y <= 5.2e+81) {
		tmp = 1.0 + (x - (y * z));
	} else {
		tmp = x * (1.0 + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-9.4d+23)) then
        tmp = x + 1.0d0
    else if (y <= 5.2d+81) then
        tmp = 1.0d0 + (x - (y * z))
    else
        tmp = x * (1.0d0 + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -9.4e+23) {
		tmp = x + 1.0;
	} else if (y <= 5.2e+81) {
		tmp = 1.0 + (x - (y * z));
	} else {
		tmp = x * (1.0 + (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -9.4e+23:
		tmp = x + 1.0
	elif y <= 5.2e+81:
		tmp = 1.0 + (x - (y * z))
	else:
		tmp = x * (1.0 + (1.0 / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -9.4e+23)
		tmp = Float64(x + 1.0);
	elseif (y <= 5.2e+81)
		tmp = Float64(1.0 + Float64(x - Float64(y * z)));
	else
		tmp = Float64(x * Float64(1.0 + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -9.4e+23)
		tmp = x + 1.0;
	elseif (y <= 5.2e+81)
		tmp = 1.0 + (x - (y * z));
	else
		tmp = x * (1.0 + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -9.4e+23], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 5.2e+81], N[(1.0 + N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.3999999999999994e23

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. sin-neg 99.8%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 45.6%

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

   if -9.3999999999999994e23 < y < 5.19999999999999984e81

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-neg-out 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. sin-neg 100.0%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 93.2%

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

      1. mul-1-neg 93.2%

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

      2. unsub-neg 93.2%

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

   7. Simplified 93.2%

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

   if 5.19999999999999984e81 < y

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-lft-neg-out 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. sin-neg 99.8%

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

      8. fma-define 99.8%

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

      9. sin-neg 99.8%

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

   3. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-undefine 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. associate-+l+ 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. add-cube-cbrt 99.2%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}\right)} + \left(x + \cos y\right) \]

      7. associate-*r* 99.3%

         \[\leadsto \color{blue}{\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)\right) \cdot \sqrt[3]{\sin y}} + \left(x + \cos y\right) \]

      8. fma-define 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right), \sqrt[3]{\sin y}, x + \cos y\right)} \]

      9. pow2 99.3%

         \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot \color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y}, x + \cos y\right) \]

      10. +-commutative 99.3%

          \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, \color{blue}{\cos y + x}\right) \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, \cos y + x\right)} \]

   7. Taylor expanded in y around 0 28.2%

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

      1. +-commutative 28.2%

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

   9. Simplified 28.2%

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

   10. Taylor expanded in x around inf 28.2%

       \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{+23}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+81}:\\ \;\;\;\;1 + \left(x - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.6% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-10}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-31}:\\ \;\;\;\;1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e-10)
   (+ x 1.0)
   (if (<= x 1.1e-31) (- 1.0 (* y z)) (* x (+ 1.0 (/ 1.0 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-10) {
		tmp = x + 1.0;
	} else if (x <= 1.1e-31) {
		tmp = 1.0 - (y * z);
	} else {
		tmp = x * (1.0 + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.1d-10)) then
        tmp = x + 1.0d0
    else if (x <= 1.1d-31) then
        tmp = 1.0d0 - (y * z)
    else
        tmp = x * (1.0d0 + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-10) {
		tmp = x + 1.0;
	} else if (x <= 1.1e-31) {
		tmp = 1.0 - (y * z);
	} else {
		tmp = x * (1.0 + (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.1e-10:
		tmp = x + 1.0
	elif x <= 1.1e-31:
		tmp = 1.0 - (y * z)
	else:
		tmp = x * (1.0 + (1.0 / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e-10)
		tmp = Float64(x + 1.0);
	elseif (x <= 1.1e-31)
		tmp = Float64(1.0 - Float64(y * z));
	else
		tmp = Float64(x * Float64(1.0 + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.1e-10)
		tmp = x + 1.0;
	elseif (x <= 1.1e-31)
		tmp = 1.0 - (y * z);
	else
		tmp = x * (1.0 + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.1e-10], N[(x + 1.0), $MachinePrecision], If[LessEqual[x, 1.1e-31], N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1e-10

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 80.7%

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

   if -2.1e-10 < x < 1.10000000000000005e-31

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. sub-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 54.9%

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

      1. mul-1-neg 54.9%

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

      2. *-commutative 54.9%

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

      3. unsub-neg 54.9%

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

      4. *-commutative 54.9%

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

   10. Simplified 54.9%

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

   if 1.10000000000000005e-31 < x

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-neg-out 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. sin-neg 100.0%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-undefine 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. associate-+l+ 100.0%

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

      5. distribute-lft-neg-in 100.0%

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

      6. add-cube-cbrt 99.6%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}\right)} + \left(x + \cos y\right) \]

      7. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)\right) \cdot \sqrt[3]{\sin y}} + \left(x + \cos y\right) \]

      8. fma-define 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot \left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right), \sqrt[3]{\sin y}, x + \cos y\right)} \]

      9. pow2 99.6%

         \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot \color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}, \sqrt[3]{\sin y}, x + \cos y\right) \]

      10. +-commutative 99.6%

          \[\leadsto \mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, \color{blue}{\cos y + x}\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-z\right) \cdot {\left(\sqrt[3]{\sin y}\right)}^{2}, \sqrt[3]{\sin y}, \cos y + x\right)} \]

   7. Taylor expanded in y around 0 79.0%

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

      1. +-commutative 79.0%

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

   9. Simplified 79.0%

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

   10. Taylor expanded in x around inf 79.1%

       \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-10}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-31}:\\ \;\;\;\;1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.8% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-9} \lor \neg \left(x \leq 1.2 \cdot 10^{-20}\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.25e-9) (not (<= x 1.2e-20))) (+ x 1.0) (- 1.0 (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.25e-9) || !(x <= 1.2e-20)) {
		tmp = x + 1.0;
	} else {
		tmp = 1.0 - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.25d-9)) .or. (.not. (x <= 1.2d-20))) then
        tmp = x + 1.0d0
    else
        tmp = 1.0d0 - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.25e-9) || !(x <= 1.2e-20)) {
		tmp = x + 1.0;
	} else {
		tmp = 1.0 - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.25e-9) or not (x <= 1.2e-20):
		tmp = x + 1.0
	else:
		tmp = 1.0 - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.25e-9) || !(x <= 1.2e-20))
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(1.0 - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.25e-9) || ~((x <= 1.2e-20)))
		tmp = x + 1.0;
	else
		tmp = 1.0 - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.25e-9], N[Not[LessEqual[x, 1.2e-20]], $MachinePrecision]], N[(x + 1.0), $MachinePrecision], N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e-9 or 1.19999999999999996e-20 < x

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 79.5%

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

   if -1.25e-9 < x < 1.19999999999999996e-20

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. sub-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 55.9%

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

      1. mul-1-neg 55.9%

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

      2. *-commutative 55.9%

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

      3. unsub-neg 55.9%

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

      4. *-commutative 55.9%

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

   10. Simplified 55.9%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-9} \lor \neg \left(x \leq 1.2 \cdot 10^{-20}\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.4% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.0) x (if (<= x 1.35e-15) 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.0) {
		tmp = x;
	} else if (x <= 1.35e-15) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x
    else if (x <= 1.35d-15) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.0) {
		tmp = x;
	} else if (x <= 1.35e-15) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.0:
		tmp = x
	elif x <= 1.35e-15:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.0)
		tmp = x;
	elseif (x <= 1.35e-15)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x;
	elseif (x <= 1.35e-15)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.0], x, If[LessEqual[x, 1.35e-15], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.35000000000000005e-15 < x

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-neg-out 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. sin-neg 100.0%

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

      8. fma-define 100.0%

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

      9. sin-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 80.7%

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

   if -1 < x < 1.35000000000000005e-15

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. sin-neg 99.9%

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

      8. fma-define 99.9%

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

      9. sin-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. neg-mul-1 98.9%

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

      2. sub-neg 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around 0 45.5%

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

Alternative 16: 62.4% accurate, 69.0× 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]
2. Step-by-step derivation

   1. cancel-sign-sub-inv 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+l+ 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-lft-neg-out 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. sin-neg 99.9%

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

   8. fma-define 99.9%

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

   9. sin-neg 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around 0 63.2%

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

6. Final simplification 63.2%

   \[\leadsto x + 1 \]
7. Add Preprocessing

Alternative 17: 22.0% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]
2. Step-by-step derivation

   1. cancel-sign-sub-inv 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+l+ 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-lft-neg-out 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. sin-neg 99.9%

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

   8. fma-define 99.9%

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

   9. sin-neg 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 60.6%

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

   1. neg-mul-1 60.6%

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

   2. sub-neg 60.6%

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

7. Simplified 60.6%

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

8. Taylor expanded in y around 0 24.7%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(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))))