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

Percentage Accurate: 99.9% → 99.9%

Time: 9.3s

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

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

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]

Derivation

1. Initial program 99.9%

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

Alternative 2: 70.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;x + z\\ \mathbf{elif}\;t\_0 \leq 0.4:\\ \;\;\;\;y + \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (<= t_0 -0.5) (+ x z) (if (<= t_0 0.4) (+ y (+ x z)) (+ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (z * cos(y));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = x + z;
	} else if (t_0 <= 0.4) {
		tmp = y + (x + z);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + sin(y)) + (z * cos(y))
    if (t_0 <= (-0.5d0)) then
        tmp = x + z
    else if (t_0 <= 0.4d0) then
        tmp = y + (x + z)
    else
        tmp = x + z
    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 <= -0.5) {
		tmp = x + z;
	} else if (t_0 <= 0.4) {
		tmp = y + (x + z);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.sin(y)) + (z * math.cos(y))
	tmp = 0
	if t_0 <= -0.5:
		tmp = x + z
	elif t_0 <= 0.4:
		tmp = y + (x + z)
	else:
		tmp = x + z
	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 <= -0.5)
		tmp = Float64(x + z);
	elseif (t_0 <= 0.4)
		tmp = Float64(y + Float64(x + z));
	else
		tmp = Float64(x + z);
	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 <= -0.5)
		tmp = x + z;
	elseif (t_0 <= 0.4)
		tmp = y + (x + z);
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(x + z), $MachinePrecision], If[LessEqual[t$95$0, 0.4], N[(y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.5 or 0.40000000000000002 < (+.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 69.2

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

   5. Applied rewrites 69.2%

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

   if -0.5 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 0.40000000000000002

   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 79.6

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

   5. Applied rewrites 79.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + z \cdot \cos y \leq -0.5:\\ \;\;\;\;x + z\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq 0.4:\\ \;\;\;\;y + \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 89.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2.95 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \mathsf{fma}\left(x, \frac{\sin y}{x}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2.95e+97)
     t_0
     (if (<= z 4.8e+115) (fma 1.0 z (fma x (/ (sin y) x) x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -2.95e+97) {
		tmp = t_0;
	} else if (z <= 4.8e+115) {
		tmp = fma(1.0, z, fma(x, (sin(y) / x), x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -2.95e+97)
		tmp = t_0;
	elseif (z <= 4.8e+115)
		tmp = fma(1.0, z, fma(x, Float64(sin(y) / x), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.95e+97], t$95$0, If[LessEqual[z, 4.8e+115], N[(1.0 * z + N[(x * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.95000000000000005e97 or 4.8000000000000001e115 < 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 89.0

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

   5. Applied rewrites 89.0%

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

   if -2.95000000000000005e97 < z < 4.8000000000000001e115

   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 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 95.4%

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

Alternative 5: 84.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -2.95 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-11}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-30}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -2.95e+97)
     t_0
     (if (<= z -1.1e-11)
       (+ x z)
       (if (<= z 5.7e-30)
         (+ x (sin y))
         (if (<= z 4e+115) (fma x (/ z x) x) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -2.95e+97) {
		tmp = t_0;
	} else if (z <= -1.1e-11) {
		tmp = x + z;
	} else if (z <= 5.7e-30) {
		tmp = x + sin(y);
	} else if (z <= 4e+115) {
		tmp = fma(x, (z / x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -2.95e+97)
		tmp = t_0;
	elseif (z <= -1.1e-11)
		tmp = Float64(x + z);
	elseif (z <= 5.7e-30)
		tmp = Float64(x + sin(y));
	elseif (z <= 4e+115)
		tmp = fma(x, Float64(z / x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.95e+97], t$95$0, If[LessEqual[z, -1.1e-11], N[(x + z), $MachinePrecision], If[LessEqual[z, 5.7e-30], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+115], N[(x * N[(z / x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.95000000000000005e97 or 4.0000000000000001e115 < 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 89.0

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

   5. Applied rewrites 89.0%

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

   if -2.95000000000000005e97 < z < -1.1000000000000001e-11

   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 87.1

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

   5. Applied rewrites 87.1%

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

   if -1.1000000000000001e-11 < z < 5.69999999999999977e-30

   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 94.9

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

   5. Applied rewrites 94.9%

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

   if 5.69999999999999977e-30 < z < 4.0000000000000001e115

   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 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 86.6%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{x}}, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+97}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-11}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-30}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-30}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.3e-11)
   (fma (cos y) z (+ x y))
   (if (<= z 5.7e-30)
     (+ x (sin y))
     (if (<= z 4e+115) (fma x (/ z x) x) (* z (cos y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e-11) {
		tmp = fma(cos(y), z, (x + y));
	} else if (z <= 5.7e-30) {
		tmp = x + sin(y);
	} else if (z <= 4e+115) {
		tmp = fma(x, (z / x), x);
	} else {
		tmp = z * cos(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.3e-11)
		tmp = fma(cos(y), z, Float64(x + y));
	elseif (z <= 5.7e-30)
		tmp = Float64(x + sin(y));
	elseif (z <= 4e+115)
		tmp = fma(x, Float64(z / x), x);
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.3e-11], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.7e-30], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+115], N[(x * N[(z / x), $MachinePrecision] + x), $MachinePrecision], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.3e-11

   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 82.0

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

   5. Applied rewrites 82.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 82.0

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

   7. Applied rewrites 82.0%

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

   if -1.3e-11 < z < 5.69999999999999977e-30

   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 94.9

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

   5. Applied rewrites 94.9%

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

   if 5.69999999999999977e-30 < z < 4.0000000000000001e115

   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 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 86.6%

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

   if 4.0000000000000001e115 < 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 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-30}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sin y\\ \mathbf{if}\;y \leq -0.008:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (sin y))))
   (if (<= y -0.008)
     t_0
     (if (<= y 5.6e+14)
       (fma
        (fma
         (* y y)
         (fma
          y
          (* y (fma (* y y) -0.001388888888888889 0.041666666666666664))
          -0.5)
         1.0)
        z
        (+ x y))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + sin(y);
	double tmp;
	if (y <= -0.008) {
		tmp = t_0;
	} else if (y <= 5.6e+14) {
		tmp = fma(fma((y * y), fma(y, (y * fma((y * y), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0), z, (x + y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + sin(y))
	tmp = 0.0
	if (y <= -0.008)
		tmp = t_0;
	elseif (y <= 5.6e+14)
		tmp = fma(fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0), z, Float64(x + y));
	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, -0.008], t$95$0, If[LessEqual[y, 5.6e+14], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0080000000000000002 or 5.6e14 < 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 61.2

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

   5. Applied rewrites 61.2%

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

   if -0.0080000000000000002 < y < 5.6e14

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.7

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

   10. Applied rewrites 98.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.008:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -11.0)
   (+ x z)
   (if (<= y 7.6e+17)
     (fma
      (fma
       (* y y)
       (fma
        y
        (* y (fma (* y y) -0.001388888888888889 0.041666666666666664))
        -0.5)
       1.0)
      z
      (+ x y))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -11.0) {
		tmp = x + z;
	} else if (y <= 7.6e+17) {
		tmp = fma(fma((y * y), fma(y, (y * fma((y * y), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0), z, (x + y));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -11.0)
		tmp = Float64(x + z);
	elseif (y <= 7.6e+17)
		tmp = fma(fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0), z, Float64(x + y));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -11.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 7.6e+17], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -11 or 7.6e17 < 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 41.9

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

   5. Applied rewrites 41.9%

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

   if -11 < y < 7.6e17

   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 98.6

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

   5. Applied rewrites 98.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 97.3

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

   10. Applied rewrites 97.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.041666666666666664, -0.5\right), 1\right), z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1)
   (+ x z)
   (if (<= y 6.5e+17)
     (fma (fma (* y y) (fma y (* y 0.041666666666666664) -0.5) 1.0) z (+ x y))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1) {
		tmp = x + z;
	} else if (y <= 6.5e+17) {
		tmp = fma(fma((y * y), fma(y, (y * 0.041666666666666664), -0.5), 1.0), z, (x + y));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1)
		tmp = Float64(x + z);
	elseif (y <= 6.5e+17)
		tmp = fma(fma(Float64(y * y), fma(y, Float64(y * 0.041666666666666664), -0.5), 1.0), z, Float64(x + y));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.1], N[(x + z), $MachinePrecision], If[LessEqual[y, 6.5e+17], N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.10000000000000009 or 6.5e17 < 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 41.9

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

   5. Applied rewrites 41.9%

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

   if -3.10000000000000009 < y < 6.5e17

   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 98.6

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

   5. Applied rewrites 98.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 97.2

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

   10. Applied rewrites 97.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot 0.041666666666666664, -0.5\right), 1\right), z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+20}:\\ \;\;\;\;z + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot -0.5, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -11.0)
   (+ x z)
   (if (<= y 2.6e+20) (+ z (fma y (fma y (* z -0.5) 1.0) x)) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -11.0) {
		tmp = x + z;
	} else if (y <= 2.6e+20) {
		tmp = z + fma(y, fma(y, (z * -0.5), 1.0), x);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -11.0)
		tmp = Float64(x + z);
	elseif (y <= 2.6e+20)
		tmp = Float64(z + fma(y, fma(y, Float64(z * -0.5), 1.0), x));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -11.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 2.6e+20], N[(z + N[(y * N[(y * N[(z * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -11 or 2.6e20 < 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 41.9

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

   5. Applied rewrites 41.9%

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

   if -11 < y < 2.6e20

   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 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+20}:\\ \;\;\;\;z + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot -0.5, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.8% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

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 + z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + z), $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 66.5

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

5. Applied rewrites 66.5%

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

6. Final simplification 66.5%

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

Reproduce

herbie shell --seed 2024234 
(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))))