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

Percentage Accurate: 99.9% → 99.9%

Time: 9.3s

Alternatives: 8

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: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -0.2 \lor \neg \left(t\_0 \leq 0.9\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (or (<= t_0 -0.2) (not (<= t_0 0.9)))
     (+ z x)
     (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ z x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (z * cos(y));
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 0.9)) {
		tmp = z + x;
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	tmp = 0.0
	if ((t_0 <= -0.2) || !(t_0 <= 0.9))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 0.9]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.20000000000000001 or 0.900000000000000022 < (+.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 72.6

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

   5. Applied rewrites 72.6%

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

   if -0.20000000000000001 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 0.900000000000000022

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + z \cdot \cos y \leq -0.2 \lor \neg \left(\left(x + \sin y\right) + z \cdot \cos y \leq 0.9\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+59} \lor \neg \left(z \leq 1.9 \cdot 10^{+56}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.8e+59) (not (<= z 1.9e+56)))
   (fma (cos y) z (+ x y))
   (fma 1.0 z (+ (sin y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.8e+59) || !(z <= 1.9e+56)) {
		tmp = fma(cos(y), z, (x + y));
	} else {
		tmp = fma(1.0, z, (sin(y) + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.8e+59) || !(z <= 1.9e+56))
		tmp = fma(cos(y), z, Float64(x + y));
	else
		tmp = fma(1.0, z, Float64(sin(y) + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.8e+59], N[Not[LessEqual[z, 1.9e+56]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8000000000000001e59 or 1.89999999999999998e56 < z

   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 94.7

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

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\left(y + x\right)} + z \cdot \cos y \]
   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. *-lft-identity N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 94.7

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

      8. lift-*.f64 N/A

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

      9. *-lft-identity 94.7

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

   7. Applied rewrites 94.7%

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

   if -3.8000000000000001e59 < z < 1.89999999999999998e56

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

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

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

      3. sin-sum N/A

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

      4. flip3-+ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 96.0%

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

   9. Applied rewrites 96.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+59} \lor \neg \left(z \leq 1.9 \cdot 10^{+56}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-40} \lor \neg \left(x \leq 1.55 \cdot 10^{-46}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\sin y + z \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.3e-40) (not (<= x 1.55e-46))) (+ z x) (+ (sin y) (* z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.3e-40) || !(x <= 1.55e-46)) {
		tmp = z + x;
	} else {
		tmp = sin(y) + (z * 1.0);
	}
	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) :: tmp
    if ((x <= (-2.3d-40)) .or. (.not. (x <= 1.55d-46))) then
        tmp = z + x
    else
        tmp = sin(y) + (z * 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.3e-40) || !(x <= 1.55e-46)) {
		tmp = z + x;
	} else {
		tmp = Math.sin(y) + (z * 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.3e-40) or not (x <= 1.55e-46):
		tmp = z + x
	else:
		tmp = math.sin(y) + (z * 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.3e-40) || !(x <= 1.55e-46))
		tmp = Float64(z + x);
	else
		tmp = Float64(sin(y) + Float64(z * 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.3e-40) || ~((x <= 1.55e-46)))
		tmp = z + x;
	else
		tmp = sin(y) + (z * 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.3e-40], N[Not[LessEqual[x, 1.55e-46]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] + N[(z * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3e-40 or 1.55e-46 < x

   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 84.9

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

   5. Applied rewrites 84.9%

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

   if -2.3e-40 < x < 1.55e-46

   1. Initial program 99.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. sin-sum N/A

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

      4. flip3-+ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 76.1%

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

   8. Taylor expanded in x around 0

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

      1. lower-sin.f64 74.7

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

   10. Applied rewrites 74.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-40} \lor \neg \left(x \leq 1.55 \cdot 10^{-46}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\sin y + z \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, z, \sin y + x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma 1.0 z (+ (sin y) x)))

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $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-cos.f64 N/A

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

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

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

   3. sin-sum N/A

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

   4. flip3-+ N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 83.0%

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

9. Applied rewrites 83.0%

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

Alternative 6: 70.6% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8000000000 \lor \neg \left(y \leq 8.5 \cdot 10^{-17}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8000000000.0) (not (<= y 8.5e-17))) (+ z x) (+ (+ z y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8000000000.0) || !(y <= 8.5e-17)) {
		tmp = z + x;
	} else {
		tmp = (z + y) + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-8000000000.0d0)) .or. (.not. (y <= 8.5d-17))) then
        tmp = z + x
    else
        tmp = (z + y) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8000000000.0) || !(y <= 8.5e-17)) {
		tmp = z + x;
	} else {
		tmp = (z + y) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8000000000.0) or not (y <= 8.5e-17):
		tmp = z + x
	else:
		tmp = (z + y) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8000000000.0) || !(y <= 8.5e-17))
		tmp = Float64(z + x);
	else
		tmp = Float64(Float64(z + y) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8000000000.0) || ~((y <= 8.5e-17)))
		tmp = z + x;
	else
		tmp = (z + y) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8000000000.0], N[Not[LessEqual[y, 8.5e-17]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(z + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8e9 or 8.5e-17 < 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 47.2

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

   5. Applied rewrites 47.2%

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

   if -8e9 < y < 8.5e-17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8000000000 \lor \neg \left(y \leq 8.5 \cdot 10^{-17}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-69} \lor \neg \left(x \leq 1.45 \cdot 10^{-65}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;z + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1e-69) (not (<= x 1.45e-65))) (+ z x) (+ z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1e-69) || !(x <= 1.45e-65)) {
		tmp = z + x;
	} else {
		tmp = z + y;
	}
	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 <= (-1d-69)) .or. (.not. (x <= 1.45d-65))) then
        tmp = z + x
    else
        tmp = z + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1e-69) || !(x <= 1.45e-65)) {
		tmp = z + x;
	} else {
		tmp = z + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1e-69) or not (x <= 1.45e-65):
		tmp = z + x
	else:
		tmp = z + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1e-69) || !(x <= 1.45e-65))
		tmp = Float64(z + x);
	else
		tmp = Float64(z + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1e-69) || ~((x <= 1.45e-65)))
		tmp = z + x;
	else
		tmp = z + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1e-69], N[Not[LessEqual[x, 1.45e-65]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999996e-70 or 1.4499999999999999e-65 < x

   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 83.2

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

   5. Applied rewrites 83.2%

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

   if -9.9999999999999996e-70 < x < 1.4499999999999999e-65

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 54.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-69} \lor \neg \left(x \leq 1.45 \cdot 10^{-65}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;z + y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.2% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-sin.f64 59.6

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

5. Applied rewrites 59.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 32.3%

   \[\leadsto z + \color{blue}{y} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(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))))