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

Percentage Accurate: 99.9% → 99.9%

Time: 12.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. lower-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 70.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -0.16 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-71}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (or (<= t_0 -0.16) (not (<= t_0 5e-71))) (+ z x) (+ (+ z y) x))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (z * cos(y));
	double tmp;
	if ((t_0 <= -0.16) || !(t_0 <= 5e-71)) {
		tmp = z + x;
	} else {
		tmp = (z + 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) :: t_0
    real(8) :: tmp
    t_0 = (x + sin(y)) + (z * cos(y))
    if ((t_0 <= (-0.16d0)) .or. (.not. (t_0 <= 5d-71))) then
        tmp = z + x
    else
        tmp = (z + y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + Math.sin(y)) + (z * Math.cos(y));
	double tmp;
	if ((t_0 <= -0.16) || !(t_0 <= 5e-71)) {
		tmp = z + x;
	} else {
		tmp = (z + y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.sin(y)) + (z * math.cos(y))
	tmp = 0
	if (t_0 <= -0.16) or not (t_0 <= 5e-71):
		tmp = z + x
	else:
		tmp = (z + y) + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	tmp = 0.0
	if ((t_0 <= -0.16) || !(t_0 <= 5e-71))
		tmp = Float64(z + x);
	else
		tmp = Float64(Float64(z + y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + sin(y)) + (z * cos(y));
	tmp = 0.0;
	if ((t_0 <= -0.16) || ~((t_0 <= 5e-71)))
		tmp = z + x;
	else
		tmp = (z + y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.16], N[Not[LessEqual[t$95$0, 5e-71]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(z + y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.160000000000000003 or 4.99999999999999998e-71 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if -0.160000000000000003 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 4.99999999999999998e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + z \cdot \cos y \leq -0.16 \lor \neg \left(\left(x + \sin y\right) + z \cdot \cos y \leq 5 \cdot 10^{-71}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+84}:\\ \;\;\;\;\left(y + x\right) + z \cdot \cos y\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-87}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\sin y + x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.8e+84)
   (+ (+ y x) (* z (cos y)))
   (if (<= z -7.6e-87)
     (+ z x)
     (if (<= z 1.8e-43)
       (+ (sin y) x)
       (if (<= z 6.2e+31) (+ z x) (* (cos y) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.8e+84) {
		tmp = (y + x) + (z * cos(y));
	} else if (z <= -7.6e-87) {
		tmp = z + x;
	} else if (z <= 1.8e-43) {
		tmp = sin(y) + x;
	} else if (z <= 6.2e+31) {
		tmp = z + x;
	} else {
		tmp = cos(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 (z <= (-5.8d+84)) then
        tmp = (y + x) + (z * cos(y))
    else if (z <= (-7.6d-87)) then
        tmp = z + x
    else if (z <= 1.8d-43) then
        tmp = sin(y) + x
    else if (z <= 6.2d+31) then
        tmp = z + x
    else
        tmp = cos(y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.8e+84) {
		tmp = (y + x) + (z * Math.cos(y));
	} else if (z <= -7.6e-87) {
		tmp = z + x;
	} else if (z <= 1.8e-43) {
		tmp = Math.sin(y) + x;
	} else if (z <= 6.2e+31) {
		tmp = z + x;
	} else {
		tmp = Math.cos(y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.8e+84:
		tmp = (y + x) + (z * math.cos(y))
	elif z <= -7.6e-87:
		tmp = z + x
	elif z <= 1.8e-43:
		tmp = math.sin(y) + x
	elif z <= 6.2e+31:
		tmp = z + x
	else:
		tmp = math.cos(y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.8e+84)
		tmp = Float64(Float64(y + x) + Float64(z * cos(y)));
	elseif (z <= -7.6e-87)
		tmp = Float64(z + x);
	elseif (z <= 1.8e-43)
		tmp = Float64(sin(y) + x);
	elseif (z <= 6.2e+31)
		tmp = Float64(z + x);
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.8e+84)
		tmp = (y + x) + (z * cos(y));
	elseif (z <= -7.6e-87)
		tmp = z + x;
	elseif (z <= 1.8e-43)
		tmp = sin(y) + x;
	elseif (z <= 6.2e+31)
		tmp = z + x;
	else
		tmp = cos(y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.8e+84], N[(N[(y + x), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -7.6e-87], N[(z + x), $MachinePrecision], If[LessEqual[z, 1.8e-43], N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 6.2e+31], N[(z + x), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.79999999999999977e84

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -5.79999999999999977e84 < z < -7.6e-87 or 1.7999999999999999e-43 < z < 6.2000000000000004e31

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if -7.6e-87 < z < 1.7999999999999999e-43

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 91.9

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

   7. Applied rewrites 91.9%

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

   if 6.2000000000000004e31 < z

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      4. flip3-+ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x + \sin y\right) + \color{blue}{z \cdot \frac{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{{\left(0 \cdot \sin y\right)}^{3}} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{{\color{blue}{\left(0 \cdot \sin y\right)}}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      6. mul0-lft N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{{\color{blue}{0}}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{0} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      8. +-lft-identity N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{{\left(1 \cdot \cos y\right)}^{3}}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

   6. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + \frac{{\cos y}^{3} \cdot z}{{\cos y}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{\cos y}^{3} \cdot z}{{\cos y}^{2}} + \left(x + \sin y\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\cos y}^{3} \cdot z}{{\cos y}^{2}}} + \left(x + \sin y\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\cos y}^{3} \cdot z}}{{\cos y}^{2}} + \left(x + \sin y\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{{\cos y}^{3} \cdot z}{\color{blue}{{\cos y}^{2}}} + \left(x + \sin y\right) \]

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow1 N/A

         \[\leadsto \frac{{\cos y}^{3}}{\color{blue}{{\cos y}^{1}}} \cdot \frac{z}{\cos y} + \left(x + \sin y\right) \]

      10. pow-div N/A

          \[\leadsto \color{blue}{{\cos y}^{\left(3 - 1\right)}} \cdot \frac{z}{\cos y} + \left(x + \sin y\right) \]

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 86.4

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

   11. Applied rewrites 86.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+84}:\\ \;\;\;\;\left(y + x\right) + z \cdot \cos y\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-87}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\sin y + x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-87}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\sin y + x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -6.8e+77)
     t_0
     (if (<= z -7.6e-87)
       (+ z x)
       (if (<= z 1.8e-43) (+ (sin y) x) (if (<= z 6.2e+31) (+ z x) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -6.8e+77) {
		tmp = t_0;
	} else if (z <= -7.6e-87) {
		tmp = z + x;
	} else if (z <= 1.8e-43) {
		tmp = sin(y) + x;
	} else if (z <= 6.2e+31) {
		tmp = z + x;
	} else {
		tmp = 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 = cos(y) * z
    if (z <= (-6.8d+77)) then
        tmp = t_0
    else if (z <= (-7.6d-87)) then
        tmp = z + x
    else if (z <= 1.8d-43) then
        tmp = sin(y) + x
    else if (z <= 6.2d+31) then
        tmp = z + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * z;
	double tmp;
	if (z <= -6.8e+77) {
		tmp = t_0;
	} else if (z <= -7.6e-87) {
		tmp = z + x;
	} else if (z <= 1.8e-43) {
		tmp = Math.sin(y) + x;
	} else if (z <= 6.2e+31) {
		tmp = z + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * z
	tmp = 0
	if z <= -6.8e+77:
		tmp = t_0
	elif z <= -7.6e-87:
		tmp = z + x
	elif z <= 1.8e-43:
		tmp = math.sin(y) + x
	elif z <= 6.2e+31:
		tmp = z + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -6.8e+77)
		tmp = t_0;
	elseif (z <= -7.6e-87)
		tmp = Float64(z + x);
	elseif (z <= 1.8e-43)
		tmp = Float64(sin(y) + x);
	elseif (z <= 6.2e+31)
		tmp = Float64(z + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * z;
	tmp = 0.0;
	if (z <= -6.8e+77)
		tmp = t_0;
	elseif (z <= -7.6e-87)
		tmp = z + x;
	elseif (z <= 1.8e-43)
		tmp = sin(y) + x;
	elseif (z <= 6.2e+31)
		tmp = z + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -6.8e+77], t$95$0, If[LessEqual[z, -7.6e-87], N[(z + x), $MachinePrecision], If[LessEqual[z, 1.8e-43], N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 6.2e+31], N[(z + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.79999999999999993e77 or 6.2000000000000004e31 < z

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      4. flip3-+ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x + \sin y\right) + \color{blue}{z \cdot \frac{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{{\left(0 \cdot \sin y\right)}^{3}} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{{\color{blue}{\left(0 \cdot \sin y\right)}}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      6. mul0-lft N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{{\color{blue}{0}}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{0} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      8. +-lft-identity N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{{\left(1 \cdot \cos y\right)}^{3}}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

   6. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + \frac{{\cos y}^{3} \cdot z}{{\cos y}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{\cos y}^{3} \cdot z}{{\cos y}^{2}} + \left(x + \sin y\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\cos y}^{3} \cdot z}{{\cos y}^{2}}} + \left(x + \sin y\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\cos y}^{3} \cdot z}}{{\cos y}^{2}} + \left(x + \sin y\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{{\cos y}^{3} \cdot z}{\color{blue}{{\cos y}^{2}}} + \left(x + \sin y\right) \]

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow1 N/A

         \[\leadsto \frac{{\cos y}^{3}}{\color{blue}{{\cos y}^{1}}} \cdot \frac{z}{\cos y} + \left(x + \sin y\right) \]

      10. pow-div N/A

          \[\leadsto \color{blue}{{\cos y}^{\left(3 - 1\right)}} \cdot \frac{z}{\cos y} + \left(x + \sin y\right) \]

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 86.0

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

   11. Applied rewrites 86.0%

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

   if -6.79999999999999993e77 < z < -7.6e-87 or 1.7999999999999999e-43 < z < 6.2000000000000004e31

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if -7.6e-87 < z < 1.7999999999999999e-43

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 91.9

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

   7. Applied rewrites 91.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+77}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-87}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\sin y + x\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+84}:\\ \;\;\;\;\left(y + x\right) + z \cdot \cos y\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;\left(x + \sin y\right) + z \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.8e+84)
   (+ (+ y x) (* z (cos y)))
   (if (<= z 6.2e+31) (+ (+ x (sin y)) (* z 1.0)) (* (cos y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.8e+84) {
		tmp = (y + x) + (z * cos(y));
	} else if (z <= 6.2e+31) {
		tmp = (x + sin(y)) + (z * 1.0);
	} else {
		tmp = cos(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 (z <= (-5.8d+84)) then
        tmp = (y + x) + (z * cos(y))
    else if (z <= 6.2d+31) then
        tmp = (x + sin(y)) + (z * 1.0d0)
    else
        tmp = cos(y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.8e+84) {
		tmp = (y + x) + (z * Math.cos(y));
	} else if (z <= 6.2e+31) {
		tmp = (x + Math.sin(y)) + (z * 1.0);
	} else {
		tmp = Math.cos(y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.8e+84:
		tmp = (y + x) + (z * math.cos(y))
	elif z <= 6.2e+31:
		tmp = (x + math.sin(y)) + (z * 1.0)
	else:
		tmp = math.cos(y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.8e+84)
		tmp = Float64(Float64(y + x) + Float64(z * cos(y)));
	elseif (z <= 6.2e+31)
		tmp = Float64(Float64(x + sin(y)) + Float64(z * 1.0));
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.8e+84)
		tmp = (y + x) + (z * cos(y));
	elseif (z <= 6.2e+31)
		tmp = (x + sin(y)) + (z * 1.0);
	else
		tmp = cos(y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.8e+84], N[(N[(y + x), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.2e+31], N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.79999999999999977e84

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -5.79999999999999977e84 < z < 6.2000000000000004e31

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 95.0%

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

   if 6.2000000000000004e31 < z

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\left(\sin y \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos y \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      4. flip3-+ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x + \sin y\right) + \color{blue}{z \cdot \frac{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \color{blue}{\frac{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{{\left(0 \cdot \sin y\right)}^{3} + {\left(1 \cdot \cos y\right)}^{3}}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{{\left(0 \cdot \sin y\right)}^{3}} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{{\color{blue}{\left(0 \cdot \sin y\right)}}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      6. mul0-lft N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{{\color{blue}{0}}^{3} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{0} + {\left(1 \cdot \cos y\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

      8. +-lft-identity N/A

         \[\leadsto \left(x + \sin y\right) + z \cdot \frac{\color{blue}{{\left(1 \cdot \cos y\right)}^{3}}}{\mathsf{fma}\left(0 \cdot \sin y, 0 \cdot \sin y, \left(1 \cdot \cos y\right) \cdot \left(1 \cdot \cos y\right) - \left(0 \cdot \sin y\right) \cdot \left(1 \cdot \cos y\right)\right)} \]

   6. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \sin y\right) + \frac{{\cos y}^{3} \cdot z}{{\cos y}^{2}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{\cos y}^{3} \cdot z}{{\cos y}^{2}} + \left(x + \sin y\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\cos y}^{3} \cdot z}{{\cos y}^{2}}} + \left(x + \sin y\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\cos y}^{3} \cdot z}}{{\cos y}^{2}} + \left(x + \sin y\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{{\cos y}^{3} \cdot z}{\color{blue}{{\cos y}^{2}}} + \left(x + \sin y\right) \]

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow1 N/A

         \[\leadsto \frac{{\cos y}^{3}}{\color{blue}{{\cos y}^{1}}} \cdot \frac{z}{\cos y} + \left(x + \sin y\right) \]

      10. pow-div N/A

          \[\leadsto \color{blue}{{\cos y}^{\left(3 - 1\right)}} \cdot \frac{z}{\cos y} + \left(x + \sin y\right) \]

      11. metadata-eval N/A

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

      12. lift-pow.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 86.4

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

   11. Applied rewrites 86.4%

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

Alternative 6: 81.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15000000000000 \lor \neg \left(y \leq 0.165\right):\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -15000000000000.0) (not (<= y 0.165)))
   (+ (sin y) x)
   (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -15000000000000.0) || !(y <= 0.165)) {
		tmp = sin(y) + x;
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -15000000000000.0) || !(y <= 0.165))
		tmp = Float64(sin(y) + x);
	else
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -15000000000000.0], N[Not[LessEqual[y, 0.165]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5e13 or 0.165000000000000008 < y

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 56.9

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

   7. Applied rewrites 56.9%

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

   if -1.5e13 < y < 0.165000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15000000000000 \lor \neg \left(y \leq 0.165\right):\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -33000000000000 \lor \neg \left(y \leq 7500000000000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -33000000000000.0) (not (<= y 7500000000000.0)))
   (+ z x)
   (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -33000000000000.0) || !(y <= 7500000000000.0)) {
		tmp = z + x;
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -33000000000000.0) || !(y <= 7500000000000.0))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -33000000000000.0], N[Not[LessEqual[y, 7500000000000.0]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e13 or 7.5e12 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 40.4

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

   5. Applied rewrites 40.4%

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

   if -3.3e13 < y < 7.5e12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.2

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

   5. Applied rewrites 96.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -33000000000000 \lor \neg \left(y \leq 7500000000000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -34000000000000 \lor \neg \left(y \leq 5 \cdot 10^{+15}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -34000000000000.0) (not (<= y 5e+15)))
   (+ z x)
   (fma (fma (* z y) -0.5 1.0) y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -34000000000000.0) || !(y <= 5e+15)) {
		tmp = z + x;
	} else {
		tmp = fma(fma((z * y), -0.5, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -34000000000000.0) || !(y <= 5e+15))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(Float64(z * y), -0.5, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -34000000000000.0], N[Not[LessEqual[y, 5e+15]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e13 or 5e15 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 40.4

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

   5. Applied rewrites 40.4%

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

   if -3.4e13 < y < 5e15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -34000000000000 \lor \neg \left(y \leq 5 \cdot 10^{+15}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -245 \lor \neg \left(y \leq 8200000000000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -245.0) (not (<= y 8200000000000.0)))
   (+ z x)
   (fma (fma (* -0.16666666666666666 y) y 1.0) y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -245.0) || !(y <= 8200000000000.0)) {
		tmp = z + x;
	} else {
		tmp = fma(fma((-0.16666666666666666 * y), y, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -245.0) || !(y <= 8200000000000.0))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(Float64(-0.16666666666666666 * y), y, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -245.0], N[Not[LessEqual[y, 8200000000000.0]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -245 or 8.2e12 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 40.3

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

   5. Applied rewrites 40.3%

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

   if -245 < y < 8.2e12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 96.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -245 \lor \neg \left(y \leq 8200000000000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.4% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 67.7

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

5. Applied rewrites 67.7%

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

6. Final simplification 67.7%

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

Alternative 11: 29.8% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-sin.f64 61.4

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

5. Applied rewrites 61.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 32.8%

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

9. Final simplification 32.8%

   \[\leadsto z + y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C"
  :precision binary64
  (+ (+ x (sin y)) (* z (cos y))))