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

Percentage Accurate: 99.9% → 99.9%

Time: 10.2s

Alternatives: 15

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

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

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

Wolfram

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 81.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + \cos y \cdot z\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;z + x\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;y + \left(z + x\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* (cos y) z))))
   (if (<= t_0 -10.0)
     (+ z x)
     (if (<= t_0 -0.01)
       (sin y)
       (if (<= t_0 1e-5) (+ y (+ z x)) (if (<= t_0 1.0) (sin y) (+ z x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (cos(y) * z);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = z + x;
	} else if (t_0 <= -0.01) {
		tmp = sin(y);
	} else if (t_0 <= 1e-5) {
		tmp = y + (z + x);
	} else if (t_0 <= 1.0) {
		tmp = sin(y);
	} else {
		tmp = z + 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)) + (cos(y) * z)
    if (t_0 <= (-10.0d0)) then
        tmp = z + x
    else if (t_0 <= (-0.01d0)) then
        tmp = sin(y)
    else if (t_0 <= 1d-5) then
        tmp = y + (z + x)
    else if (t_0 <= 1.0d0) then
        tmp = sin(y)
    else
        tmp = z + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + Math.sin(y)) + (Math.cos(y) * z);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = z + x;
	} else if (t_0 <= -0.01) {
		tmp = Math.sin(y);
	} else if (t_0 <= 1e-5) {
		tmp = y + (z + x);
	} else if (t_0 <= 1.0) {
		tmp = Math.sin(y);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.sin(y)) + (math.cos(y) * z)
	tmp = 0
	if t_0 <= -10.0:
		tmp = z + x
	elif t_0 <= -0.01:
		tmp = math.sin(y)
	elif t_0 <= 1e-5:
		tmp = y + (z + x)
	elif t_0 <= 1.0:
		tmp = math.sin(y)
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(cos(y) * z))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = Float64(z + x);
	elseif (t_0 <= -0.01)
		tmp = sin(y);
	elseif (t_0 <= 1e-5)
		tmp = Float64(y + Float64(z + x));
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + sin(y)) + (cos(y) * z);
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = z + x;
	elseif (t_0 <= -0.01)
		tmp = sin(y);
	elseif (t_0 <= 1e-5)
		tmp = y + (z + x);
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[(z + x), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 1e-5], N[(y + N[(z + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[y], $MachinePrecision], N[(z + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -10 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   2. 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 76.1

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

   5. Applied rewrites 76.1%

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

   if -10 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.0100000000000000002 or 1.00000000000000008e-5 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   4. 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 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.4%

      \[\leadsto \sin y \]

   if -0.0100000000000000002 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1.00000000000000008e-5

   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. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + \cos y \cdot z \leq -10:\\ \;\;\;\;z + x\\ \mathbf{elif}\;\left(x + \sin y\right) + \cos y \cdot z \leq -0.01:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;\left(x + \sin y\right) + \cos y \cdot z \leq 10^{-5}:\\ \;\;\;\;y + \left(z + x\right)\\ \mathbf{elif}\;\left(x + \sin y\right) + \cos y \cdot z \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-47}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 1700:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -1.55e+129)
     t_0
     (if (<= z -1.4e-47) (+ z x) (if (<= z 1700.0) (+ x (sin y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -1.55e+129) {
		tmp = t_0;
	} else if (z <= -1.4e-47) {
		tmp = z + x;
	} else if (z <= 1700.0) {
		tmp = x + sin(y);
	} 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 <= (-1.55d+129)) then
        tmp = t_0
    else if (z <= (-1.4d-47)) then
        tmp = z + x
    else if (z <= 1700.0d0) then
        tmp = x + sin(y)
    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 <= -1.55e+129) {
		tmp = t_0;
	} else if (z <= -1.4e-47) {
		tmp = z + x;
	} else if (z <= 1700.0) {
		tmp = x + Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * z
	tmp = 0
	if z <= -1.55e+129:
		tmp = t_0
	elif z <= -1.4e-47:
		tmp = z + x
	elif z <= 1700.0:
		tmp = x + math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -1.55e+129)
		tmp = t_0;
	elseif (z <= -1.4e-47)
		tmp = Float64(z + x);
	elseif (z <= 1700.0)
		tmp = Float64(x + sin(y));
	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 <= -1.55e+129)
		tmp = t_0;
	elseif (z <= -1.4e-47)
		tmp = z + x;
	elseif (z <= 1700.0)
		tmp = x + sin(y);
	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, -1.55e+129], t$95$0, If[LessEqual[z, -1.4e-47], N[(z + x), $MachinePrecision], If[LessEqual[z, 1700.0], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55e129 or 1700 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -1.55e129 < z < -1.39999999999999996e-47

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 81.1

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

   5. Applied rewrites 81.1%

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

   if -1.39999999999999996e-47 < z < 1700

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   4. 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 91.2

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

   5. Applied rewrites 91.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+129}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-47}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 1700:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, \cos y, x\right)\\ \mathbf{if}\;z \leq -50000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.8:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (cos y) x)))
   (if (<= z -50000000.0) t_0 (if (<= z 2.8) (fma 1.0 z (+ x (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, cos(y), x);
	double tmp;
	if (z <= -50000000.0) {
		tmp = t_0;
	} else if (z <= 2.8) {
		tmp = fma(1.0, z, (x + sin(y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, cos(y), x)
	tmp = 0.0
	if (z <= -50000000.0)
		tmp = t_0;
	elseif (z <= 2.8)
		tmp = fma(1.0, z, Float64(x + sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -50000000.0], t$95$0, If[LessEqual[z, 2.8], N[(1.0 * z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5e7 or 2.7999999999999998 < z

   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 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sin.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 99.5%

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

   if -5e7 < z < 2.7999999999999998

   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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.8%

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

Alternative 5: 95.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, \cos y, x\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-11}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (cos y) x)))
   (if (<= z -4e-23) t_0 (if (<= z 1.25e-11) (+ x (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, cos(y), x);
	double tmp;
	if (z <= -4e-23) {
		tmp = t_0;
	} else if (z <= 1.25e-11) {
		tmp = x + sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, cos(y), x)
	tmp = 0.0
	if (z <= -4e-23)
		tmp = t_0;
	elseif (z <= 1.25e-11)
		tmp = Float64(x + sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -4e-23], t$95$0, If[LessEqual[z, 1.25e-11], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.99999999999999984e-23 or 1.25000000000000005e-11 < z

   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 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-sin.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 98.3%

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

   if -3.99999999999999984e-23 < z < 1.25000000000000005e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   4. 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 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(z, \cos y, x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-11}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \cos y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sin y\\ \mathbf{if}\;y \leq -0.014:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;\left(y + x\right) + \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), z \cdot -0.5\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (sin y))))
   (if (<= y -0.014)
     t_0
     (if (<= y 0.48)
       (+
        (+ y x)
        (fma
         (* y y)
         (fma
          (* y y)
          (* z (fma y (* y -0.001388888888888889) 0.041666666666666664))
          (* z -0.5))
         z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + sin(y);
	double tmp;
	if (y <= -0.014) {
		tmp = t_0;
	} else if (y <= 0.48) {
		tmp = (y + x) + fma((y * y), fma((y * y), (z * fma(y, (y * -0.001388888888888889), 0.041666666666666664)), (z * -0.5)), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + sin(y))
	tmp = 0.0
	if (y <= -0.014)
		tmp = t_0;
	elseif (y <= 0.48)
		tmp = Float64(Float64(y + x) + fma(Float64(y * y), fma(Float64(y * y), Float64(z * fma(y, Float64(y * -0.001388888888888889), 0.041666666666666664)), Float64(z * -0.5)), z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.014], t$95$0, If[LessEqual[y, 0.48], N[(N[(y + x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(z * N[(y * N[(y * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(z * -0.5), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0140000000000000003 or 0.47999999999999998 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   4. 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 62.7

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

   5. Applied rewrites 62.7%

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

   if -0.0140000000000000003 < y < 0.47999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y + x\right) + \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y + x\right) + \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right) + z\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(y + x\right) + \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right), z\right)} \]

   8. Applied rewrites 99.7%

      \[\leadsto \left(y + x\right) + \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), z \cdot -0.5\right), z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.014:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;\left(y + x\right) + \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), z \cdot -0.5\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -62:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 1600:\\ \;\;\;\;\left(y + x\right) + \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), z \cdot -0.5\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -62.0)
   (+ z x)
   (if (<= y 1600.0)
     (+
      (+ y x)
      (fma
       (* y y)
       (fma
        (* y y)
        (* z (fma y (* y -0.001388888888888889) 0.041666666666666664))
        (* z -0.5))
       z))
     (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -62.0) {
		tmp = z + x;
	} else if (y <= 1600.0) {
		tmp = (y + x) + fma((y * y), fma((y * y), (z * fma(y, (y * -0.001388888888888889), 0.041666666666666664)), (z * -0.5)), z);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -62.0)
		tmp = Float64(z + x);
	elseif (y <= 1600.0)
		tmp = Float64(Float64(y + x) + fma(Float64(y * y), fma(Float64(y * y), Float64(z * fma(y, Float64(y * -0.001388888888888889), 0.041666666666666664)), Float64(z * -0.5)), z));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -62.0], N[(z + x), $MachinePrecision], If[LessEqual[y, 1600.0], N[(N[(y + x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(z * N[(y * N[(y * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(z * -0.5), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -62 or 1600 < 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 40.6

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

   5. Applied rewrites 40.6%

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

   if -62 < y < 1600

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y + x\right) + \color{blue}{\left(z + {y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y + x\right) + \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right)\right) + z\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(y + x\right) + \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{2} \cdot z + {y}^{2} \cdot \left(\frac{-1}{720} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{24} \cdot z\right), z\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \left(y + x\right) + \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), z \cdot -0.5\right), z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -62:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 1600:\\ \;\;\;\;\left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -62.0)
   (+ z x)
   (if (<= y 1600.0)
     (+
      (+ y x)
      (*
       z
       (fma
        (* y y)
        (fma
         (* y y)
         (fma y (* y -0.001388888888888889) 0.041666666666666664)
         -0.5)
        1.0)))
     (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -62.0) {
		tmp = z + x;
	} else if (y <= 1600.0) {
		tmp = (y + x) + (z * fma((y * y), fma((y * y), fma(y, (y * -0.001388888888888889), 0.041666666666666664), -0.5), 1.0));
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -62.0)
		tmp = Float64(z + x);
	elseif (y <= 1600.0)
		tmp = Float64(Float64(y + x) + Float64(z * fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(y * -0.001388888888888889), 0.041666666666666664), -0.5), 1.0)));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -62.0], N[(z + x), $MachinePrecision], If[LessEqual[y, 1600.0], N[(N[(y + x), $MachinePrecision] + N[(z * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -62 or 1600 < 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 40.6

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

   5. Applied rewrites 40.6%

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

   if -62 < y < 1600

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, {y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right)}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right), 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right), 1\right) \]

      10. +-commutative N/A

          \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \]

      11. *-commutative N/A

          \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      12. unpow2 N/A

          \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(y \cdot y\right)} \cdot \frac{-1}{720} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      13. associate-*l* N/A

          \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(y \cdot \frac{-1}{720}\right)} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \]

      15. lower-*.f64 99.2

          \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \color{blue}{y \cdot -0.001388888888888889}, 0.041666666666666664\right), -0.5\right), 1\right) \]

   8. Applied rewrites 99.2%

      \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+33}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 280:\\ \;\;\;\;\left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.5e+33)
   (+ z x)
   (if (<= y 280.0)
     (+
      (+ y x)
      (* z (fma (* y y) (fma (* y y) 0.041666666666666664 -0.5) 1.0)))
     (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.5e+33) {
		tmp = z + x;
	} else if (y <= 280.0) {
		tmp = (y + x) + (z * fma((y * y), fma((y * y), 0.041666666666666664, -0.5), 1.0));
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.5e+33)
		tmp = Float64(z + x);
	elseif (y <= 280.0)
		tmp = Float64(Float64(y + x) + Float64(z * fma(Float64(y * y), fma(Float64(y * y), 0.041666666666666664, -0.5), 1.0)));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.5e+33], N[(z + x), $MachinePrecision], If[LessEqual[y, 280.0], N[(N[(y + x), $MachinePrecision] + N[(z * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e33 or 280 < 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 41.0

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

   5. Applied rewrites 41.0%

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

   if -4.5e33 < y < 280

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{24} \cdot {y}^{2} - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} \cdot {y}^{2} - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24} \cdot {y}^{2} - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{24} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, {y}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{24}, \frac{-1}{2}\right), 1\right) \]

      10. lower-*.f64 97.0

          \[\leadsto \left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.041666666666666664, -0.5\right), 1\right) \]

   8. Applied rewrites 97.0%

      \[\leadsto \left(y + x\right) + z \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+32}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 1600:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, z \cdot -0.5\right), 1\right), z + x\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+32)
   (+ z x)
   (if (<= y 1600.0)
     (fma y (fma y (fma y -0.16666666666666666 (* z -0.5)) 1.0) (+ z x))
     (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+32) {
		tmp = z + x;
	} else if (y <= 1600.0) {
		tmp = fma(y, fma(y, fma(y, -0.16666666666666666, (z * -0.5)), 1.0), (z + x));
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.8e+32)
		tmp = Float64(z + x);
	elseif (y <= 1600.0)
		tmp = fma(y, fma(y, fma(y, -0.16666666666666666, Float64(z * -0.5)), 1.0), Float64(z + x));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.8e+32], N[(z + x), $MachinePrecision], If[LessEqual[y, 1600.0], N[(y * N[(y * N[(y * -0.16666666666666666 + N[(z * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(z + x), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7999999999999998e32 or 1600 < 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 40.7

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

   5. Applied rewrites 40.7%

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

   if -1.7999999999999998e32 < y < 1600

   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. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 11: 69.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+33}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+46}:\\ \;\;\;\;z + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot -0.5, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.8e+33)
   (+ z x)
   (if (<= y 7.5e+46) (+ z (fma y (fma y (* z -0.5) 1.0) x)) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.8e+33) {
		tmp = z + x;
	} else if (y <= 7.5e+46) {
		tmp = z + fma(y, fma(y, (z * -0.5), 1.0), x);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.8e+33)
		tmp = Float64(z + x);
	elseif (y <= 7.5e+46)
		tmp = Float64(z + fma(y, fma(y, Float64(z * -0.5), 1.0), x));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.8e+33], N[(z + x), $MachinePrecision], If[LessEqual[y, 7.5e+46], N[(z + N[(y * N[(y * N[(z * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.7999999999999999e33 or 7.5000000000000003e46 < 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 -6.7999999999999999e33 < y < 7.5000000000000003e46

   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

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

Alternative 12: 69.6% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 1200000000000:\\ \;\;\;\;y + \left(z + x\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2) (+ z x) (if (<= y 1200000000000.0) (+ y (+ z x)) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2) {
		tmp = z + x;
	} else if (y <= 1200000000000.0) {
		tmp = y + (z + x);
	} else {
		tmp = z + 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 <= (-3.2d0)) then
        tmp = z + x
    else if (y <= 1200000000000.0d0) then
        tmp = y + (z + x)
    else
        tmp = z + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2) {
		tmp = z + x;
	} else if (y <= 1200000000000.0) {
		tmp = y + (z + x);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.2:
		tmp = z + x
	elif y <= 1200000000000.0:
		tmp = y + (z + x)
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2)
		tmp = Float64(z + x);
	elseif (y <= 1200000000000.0)
		tmp = Float64(y + Float64(z + x));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.2)
		tmp = z + x;
	elseif (y <= 1200000000000.0)
		tmp = y + (z + x);
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.2], N[(z + x), $MachinePrecision], If[LessEqual[y, 1200000000000.0], N[(y + N[(z + x), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2000000000000002 or 1.2e12 < 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 41.4

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

   5. Applied rewrites 41.4%

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

   if -3.2000000000000002 < y < 1.2e12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.1

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

   5. Applied rewrites 96.1%

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

Alternative 13: 66.9% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-78}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-241}:\\ \;\;\;\;y + z\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e-78) (+ z x) (if (<= x 2.7e-241) (+ y z) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-78) {
		tmp = z + x;
	} else if (x <= 2.7e-241) {
		tmp = y + z;
	} else {
		tmp = z + 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 <= (-1.4d-78)) then
        tmp = z + x
    else if (x <= 2.7d-241) then
        tmp = y + z
    else
        tmp = z + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-78) {
		tmp = z + x;
	} else if (x <= 2.7e-241) {
		tmp = y + z;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.4e-78:
		tmp = z + x
	elif x <= 2.7e-241:
		tmp = y + z
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e-78)
		tmp = Float64(z + x);
	elseif (x <= 2.7e-241)
		tmp = Float64(y + z);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4e-78)
		tmp = z + x;
	elseif (x <= 2.7e-241)
		tmp = y + z;
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4e-78], N[(z + x), $MachinePrecision], If[LessEqual[x, 2.7e-241], N[(y + z), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.40000000000000012e-78 or 2.6999999999999999e-241 < x

   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 70.9

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

   5. Applied rewrites 70.9%

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

   if -1.40000000000000012e-78 < x < 2.6999999999999999e-241

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.2%

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

Alternative 14: 50.5% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+46}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{+47}:\\ \;\;\;\;y + z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e+46) (+ y x) (if (<= x 6.7e+47) (+ y z) (+ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+46) {
		tmp = y + x;
	} else if (x <= 6.7e+47) {
		tmp = y + z;
	} else {
		tmp = 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 (x <= (-3.1d+46)) then
        tmp = y + x
    else if (x <= 6.7d+47) then
        tmp = y + z
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+46) {
		tmp = y + x;
	} else if (x <= 6.7e+47) {
		tmp = y + z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.1e+46:
		tmp = y + x
	elif x <= 6.7e+47:
		tmp = y + z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.1e+46)
		tmp = Float64(y + x);
	elseif (x <= 6.7e+47)
		tmp = Float64(y + z);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.1e+46)
		tmp = y + x;
	elseif (x <= 6.7e+47)
		tmp = y + z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.1e+46], N[(y + x), $MachinePrecision], If[LessEqual[x, 6.7e+47], N[(y + z), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.09999999999999975e46 or 6.69999999999999973e47 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.6%

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

   if -3.09999999999999975e46 < x < 6.69999999999999973e47

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 44.0%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+46}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{+47}:\\ \;\;\;\;y + z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.1% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y + 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

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 61.0

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

5. Applied rewrites 61.0%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 36.6%

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

9. Final simplification 36.6%

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

Reproduce

herbie shell --seed 2024220 
(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))))