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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * 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-+.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

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 77.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+26}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-93}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.8e+26)
   (+ z x)
   (if (<= x -5e-93)
     (* (cos y) z)
     (if (<= x 3.7e-12) (fma 1.0 z (sin y)) (+ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e+26) {
		tmp = z + x;
	} else if (x <= -5e-93) {
		tmp = cos(y) * z;
	} else if (x <= 3.7e-12) {
		tmp = fma(1.0, z, sin(y));
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.8e+26)
		tmp = Float64(z + x);
	elseif (x <= -5e-93)
		tmp = Float64(cos(y) * z);
	elseif (x <= 3.7e-12)
		tmp = fma(1.0, z, sin(y));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.8e+26], N[(z + x), $MachinePrecision], If[LessEqual[x, -5e-93], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], If[LessEqual[x, 3.7e-12], N[(1.0 * z + N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.80000000000000009e26 or 3.69999999999999999e-12 < 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 88.9

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

   5. Applied rewrites 88.9%

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

   if -4.80000000000000009e26 < x < -4.99999999999999994e-93

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 77.8%

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

   if -4.99999999999999994e-93 < x < 3.69999999999999999e-12

   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 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(1, z, \sin y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 73.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+26}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-93}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+138} \lor \neg \left(z \leq 8.8 \cdot 10^{+69}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.2e+138) (not (<= z 8.8e+69)))
   (* (cos y) z)
   (fma 1.0 z (+ (sin y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.2e+138) || !(z <= 8.8e+69)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(1.0, z, (sin(y) + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.2e+138) || !(z <= 8.8e+69))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(1.0, z, Float64(sin(y) + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.2e+138], N[Not[LessEqual[z, 8.8e+69]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2000000000000001e138 or 8.8000000000000006e69 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 90.0%

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

   if -2.2000000000000001e138 < z < 8.8000000000000006e69

   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

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 94.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+138} \lor \neg \left(z \leq 8.8 \cdot 10^{+69}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+138}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.2e+138)
   (* (cos y) z)
   (if (<= z 1.2e+45) (fma 1.0 z (+ (sin y) x)) (fma (cos y) z (+ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e+138) {
		tmp = cos(y) * z;
	} else if (z <= 1.2e+45) {
		tmp = fma(1.0, z, (sin(y) + x));
	} else {
		tmp = fma(cos(y), z, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.2e+138)
		tmp = Float64(cos(y) * z);
	elseif (z <= 1.2e+45)
		tmp = fma(1.0, z, Float64(sin(y) + x));
	else
		tmp = fma(cos(y), z, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.2e+138], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 1.2e+45], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2000000000000001e138

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.7%

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

   if -2.2000000000000001e138 < z < 1.19999999999999995e45

   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

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 94.4%

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

   if 1.19999999999999995e45 < 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 87.4

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

   5. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\left(y + x\right)} + z \cdot \cos y \]
   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 87.4

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

   7. Applied rewrites 87.4%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+138}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+138} \lor \neg \left(z \leq 8.8 \cdot 10^{+69}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.2e+138) (not (<= z 8.8e+69))) (* (cos y) z) (+ z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.2e+138) || !(z <= 8.8e+69)) {
		tmp = cos(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 ((z <= (-2.2d+138)) .or. (.not. (z <= 8.8d+69))) then
        tmp = cos(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 ((z <= -2.2e+138) || !(z <= 8.8e+69)) {
		tmp = Math.cos(y) * z;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.2e+138) or not (z <= 8.8e+69):
		tmp = math.cos(y) * z
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.2e+138) || !(z <= 8.8e+69))
		tmp = Float64(cos(y) * z);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.2e+138) || ~((z <= 8.8e+69)))
		tmp = cos(y) * z;
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.2e+138], N[Not[LessEqual[z, 8.8e+69]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2000000000000001e138 or 8.8000000000000006e69 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 90.0%

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

   if -2.2000000000000001e138 < z < 8.8000000000000006e69

   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 70.0

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

   5. Applied rewrites 70.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+138} \lor \neg \left(z \leq 8.8 \cdot 10^{+69}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -65000 \lor \neg \left(y \leq 1.15 \cdot 10^{+28}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -65000.0) (not (<= y 1.15e+28)))
   (+ z x)
   (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -65000.0) || !(y <= 1.15e+28)) {
		tmp = z + x;
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -65000.0) || !(y <= 1.15e+28))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -65000.0], N[Not[LessEqual[y, 1.15e+28]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -65000 or 1.14999999999999992e28 < 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 37.7

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

   5. Applied rewrites 37.7%

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

   if -65000 < y < 1.14999999999999992e28

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -65000 \lor \neg \left(y \leq 1.15 \cdot 10^{+28}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.7% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 66.1

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

5. Applied rewrites 66.1%

   \[\leadsto \color{blue}{z + x} \]
6. Add Preprocessing

Alternative 8: 29.9% 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 60.4

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

5. Applied rewrites 60.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 29.4%

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

Reproduce

herbie shell --seed 2024320 
(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))))