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

Percentage Accurate: 99.9% → 99.9%

Time: 9.5s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 80.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + \cos y \cdot z\\ \mathbf{if}\;t\_0 \leq -4:\\ \;\;\;\;z + x\\ \mathbf{elif}\;t\_0 \leq -0.2:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 10^{-10}:\\ \;\;\;\;y + \left(z + x\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* (cos y) z))))
   (if (<= t_0 -4.0)
     (+ z x)
     (if (<= t_0 -0.2)
       (sin y)
       (if (<= t_0 1e-10) (+ y (+ z x)) (if (<= t_0 1.0) (sin y) (+ z x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (cos(y) * z);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = z + x;
	} else if (t_0 <= -0.2) {
		tmp = sin(y);
	} else if (t_0 <= 1e-10) {
		tmp = y + (z + x);
	} else if (t_0 <= 1.0) {
		tmp = sin(y);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + sin(y)) + (cos(y) * z)
    if (t_0 <= (-4.0d0)) then
        tmp = z + x
    else if (t_0 <= (-0.2d0)) then
        tmp = sin(y)
    else if (t_0 <= 1d-10) then
        tmp = y + (z + x)
    else if (t_0 <= 1.0d0) then
        tmp = sin(y)
    else
        tmp = z + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + Math.sin(y)) + (Math.cos(y) * z);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = z + x;
	} else if (t_0 <= -0.2) {
		tmp = Math.sin(y);
	} else if (t_0 <= 1e-10) {
		tmp = y + (z + x);
	} else if (t_0 <= 1.0) {
		tmp = Math.sin(y);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.sin(y)) + (math.cos(y) * z)
	tmp = 0
	if t_0 <= -4.0:
		tmp = z + x
	elif t_0 <= -0.2:
		tmp = math.sin(y)
	elif t_0 <= 1e-10:
		tmp = y + (z + x)
	elif t_0 <= 1.0:
		tmp = math.sin(y)
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(cos(y) * z))
	tmp = 0.0
	if (t_0 <= -4.0)
		tmp = Float64(z + x);
	elseif (t_0 <= -0.2)
		tmp = sin(y);
	elseif (t_0 <= 1e-10)
		tmp = Float64(y + Float64(z + x));
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + sin(y)) + (cos(y) * z);
	tmp = 0.0;
	if (t_0 <= -4.0)
		tmp = z + x;
	elseif (t_0 <= -0.2)
		tmp = sin(y);
	elseif (t_0 <= 1e-10)
		tmp = y + (z + x);
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4.0], N[(z + x), $MachinePrecision], If[LessEqual[t$95$0, -0.2], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 1e-10], N[(y + N[(z + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[y], $MachinePrecision], N[(z + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -4 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   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 80.4

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

   5. Applied rewrites 80.4%

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

   if -4 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.20000000000000001 or 1.00000000000000004e-10 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1

   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. flip-+ N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \sin y}{x \cdot x - \sin y \cdot \sin y}}} + z \cdot \cos y \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \sin y}{x \cdot x - \sin y \cdot \sin y}}} + z \cdot \cos y \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{x \cdot x - \sin y \cdot \sin y}{x - \sin y}}}} + z \cdot \cos y \]

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   9. 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 98.0

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

   10. Applied rewrites 98.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.0%

       \[\leadsto \sin y \]

   if -0.20000000000000001 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1.00000000000000004e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + \cos y \cdot z \leq -4:\\ \;\;\;\;z + x\\ \mathbf{elif}\;\left(x + \sin y\right) + \cos y \cdot z \leq -0.2:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;\left(x + \sin y\right) + \cos y \cdot z \leq 10^{-10}:\\ \;\;\;\;y + \left(z + x\right)\\ \mathbf{elif}\;\left(x + \sin y\right) + \cos y \cdot z \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{+174}:\\ \;\;\;\;x \cdot \left(1 + \frac{t\_0}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -3.3e+85)
     (fma (cos y) z (+ y x))
     (if (<= z 2.5e-30)
       (fma 1.0 z (+ x (sin y)))
       (if (<= z 1.52e+174) (* x (+ 1.0 (/ t_0 x))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -3.3e+85) {
		tmp = fma(cos(y), z, (y + x));
	} else if (z <= 2.5e-30) {
		tmp = fma(1.0, z, (x + sin(y)));
	} else if (z <= 1.52e+174) {
		tmp = x * (1.0 + (t_0 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -3.3e+85)
		tmp = fma(cos(y), z, Float64(y + x));
	elseif (z <= 2.5e-30)
		tmp = fma(1.0, z, Float64(x + sin(y)));
	elseif (z <= 1.52e+174)
		tmp = Float64(x * Float64(1.0 + Float64(t_0 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.3e+85], N[(N[Cos[y], $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5e-30], N[(1.0 * z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.52e+174], N[(x * N[(1.0 + N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.2999999999999999e85

   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 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\left(y + x\right)} + z \cdot \cos y \]
   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 90.4

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

   7. Applied rewrites 90.4%

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

   if -3.2999999999999999e85 < z < 2.49999999999999986e-30

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 96.5%

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

   if 2.49999999999999986e-30 < z < 1.52000000000000004e174

   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. flip-+ N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \sin y}{x \cdot x - \sin y \cdot \sin y}}} + z \cdot \cos y \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \sin y}{x \cdot x - \sin y \cdot \sin y}}} + z \cdot \cos y \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{x \cdot x - \sin y \cdot \sin y}{x - \sin y}}}} + z \cdot \cos y \]

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 57.6

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

   7. Applied rewrites 57.6%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-frac-neg2 N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 89.0%

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

   11. Taylor expanded in z around inf

       \[\leadsto x \cdot \left(1 + \frac{z \cdot \cos y}{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 88.7%

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

   if 1.52000000000000004e174 < z

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{+174}:\\ \;\;\;\;x \cdot \left(1 + \frac{\cos y \cdot z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;z \leq -4.4:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, t\_0\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ 1.0 x))))
   (if (<= z -4.4)
     (fma (cos y) z t_0)
     (if (<= z 1.02e-22) (fma 1.0 z (+ x (sin y))) (+ t_0 (* (cos y) z))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (z <= -4.4) {
		tmp = fma(cos(y), z, t_0);
	} else if (z <= 1.02e-22) {
		tmp = fma(1.0, z, (x + sin(y)));
	} else {
		tmp = t_0 + (cos(y) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (z <= -4.4)
		tmp = fma(cos(y), z, t_0);
	elseif (z <= 1.02e-22)
		tmp = fma(1.0, z, Float64(x + sin(y)));
	else
		tmp = Float64(t_0 + Float64(cos(y) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.4], N[(N[Cos[y], $MachinePrecision] * z + t$95$0), $MachinePrecision], If[LessEqual[z, 1.02e-22], N[(1.0 * z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.4000000000000004

   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. flip-+ N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \sin y}{x \cdot x - \sin y \cdot \sin y}}} + z \cdot \cos y \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \sin y}{x \cdot x - \sin y \cdot \sin y}}} + z \cdot \cos y \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{x \cdot x - \sin y \cdot \sin y}{x - \sin y}}}} + z \cdot \cos y \]

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} + z \cdot \cos y \]
   6. Step-by-step derivation

      1. lower-/.f64 99.2

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} + z \cdot \cos y \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} + z \cdot \cos y \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1}{x}} + z \cdot \cos y} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{x}} + \color{blue}{z \cdot \cos y} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{x}} + z \cdot \color{blue}{\cos y} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \cos y + \frac{1}{\frac{1}{x}}} \]

      5. lift-cos.f64 N/A

         \[\leadsto z \cdot \color{blue}{\cos y} + \frac{1}{\frac{1}{x}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} + \frac{1}{\frac{1}{x}} \]

      7. lower-fma.f64 99.2

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

   9. Applied rewrites 99.2%

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

   if -4.4000000000000004 < z < 1.02000000000000002e-22

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.4%

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

   if 1.02000000000000002e-22 < z

   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. flip-+ N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \sin y}{x \cdot x - \sin y \cdot \sin y}}} + z \cdot \cos y \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \sin y}{x \cdot x - \sin y \cdot \sin y}}} + z \cdot \cos y \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{x \cdot x - \sin y \cdot \sin y}{x - \sin y}}}} + z \cdot \cos y \]

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} + z \cdot \cos y \]
   6. Step-by-step derivation

      1. lower-/.f64 99.7

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} + z \cdot \cos y \]

   7. Applied rewrites 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \frac{1}{\frac{1}{x}}\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}} + \cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, z, \frac{1}{\frac{1}{x}}\right)\\ \mathbf{if}\;z \leq -4.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (cos y) z (/ 1.0 (/ 1.0 x)))))
   (if (<= z -4.4) t_0 (if (<= z 1.02e-22) (fma 1.0 z (+ x (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(cos(y), z, (1.0 / (1.0 / x)));
	double tmp;
	if (z <= -4.4) {
		tmp = t_0;
	} else if (z <= 1.02e-22) {
		tmp = fma(1.0, z, (x + sin(y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(cos(y), z, Float64(1.0 / Float64(1.0 / x)))
	tmp = 0.0
	if (z <= -4.4)
		tmp = t_0;
	elseif (z <= 1.02e-22)
		tmp = fma(1.0, z, Float64(x + sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z + N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.4], t$95$0, If[LessEqual[z, 1.02e-22], N[(1.0 * z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4000000000000004 or 1.02000000000000002e-22 < z

   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. flip-+ N/A

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

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \sin y}{x \cdot x - \sin y \cdot \sin y}}} + z \cdot \cos y \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \sin y}{x \cdot x - \sin y \cdot \sin y}}} + z \cdot \cos y \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{x \cdot x - \sin y \cdot \sin y}{x - \sin y}}}} + z \cdot \cos y \]

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} + z \cdot \cos y \]
   6. Step-by-step derivation

      1. lower-/.f64 99.5

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} + z \cdot \cos y \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} + z \cdot \cos y \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1}{x}} + z \cdot \cos y} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{x}} + \color{blue}{z \cdot \cos y} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{x}} + z \cdot \color{blue}{\cos y} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \cos y + \frac{1}{\frac{1}{x}}} \]

      5. lift-cos.f64 N/A

         \[\leadsto z \cdot \color{blue}{\cos y} + \frac{1}{\frac{1}{x}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\cos y \cdot z} + \frac{1}{\frac{1}{x}} \]

      7. lower-fma.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if -4.4000000000000004 < z < 1.02000000000000002e-22

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.4%

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

Alternative 6: 89.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.3e+85)
   (fma (cos y) z (+ y x))
   (if (<= z 5e+116) (fma 1.0 z (+ x (sin y))) (* (cos y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.3e+85) {
		tmp = fma(cos(y), z, (y + x));
	} else if (z <= 5e+116) {
		tmp = fma(1.0, z, (x + sin(y)));
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.3e+85)
		tmp = fma(cos(y), z, Float64(y + x));
	elseif (z <= 5e+116)
		tmp = fma(1.0, z, Float64(x + sin(y)));
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.3e+85], N[(N[Cos[y], $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e+116], N[(1.0 * z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2999999999999999e85

   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 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\left(y + x\right)} + z \cdot \cos y \]
   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 90.4

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

   7. Applied rewrites 90.4%

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

   if -3.2999999999999999e85 < z < 5.00000000000000025e116

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 93.0%

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

   if 5.00000000000000025e116 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sin y\\ \mathbf{if}\;y \leq -0.0175:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.49:\\ \;\;\;\;\left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (sin y))))
   (if (<= y -0.0175)
     t_0
     (if (<= y 0.49)
       (+
        (+ y x)
        (*
         z
         (fma
          (* y y)
          (fma
           (* y y)
           (fma y (* y -0.001388888888888889) 0.041666666666666664)
           -0.5)
          1.0)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + sin(y);
	double tmp;
	if (y <= -0.0175) {
		tmp = t_0;
	} else if (y <= 0.49) {
		tmp = (y + x) + (z * fma((y * y), fma((y * y), fma(y, (y * -0.001388888888888889), 0.041666666666666664), -0.5), 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + sin(y))
	tmp = 0.0
	if (y <= -0.0175)
		tmp = t_0;
	elseif (y <= 0.49)
		tmp = Float64(Float64(y + x) + Float64(z * fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(y * -0.001388888888888889), 0.041666666666666664), -0.5), 1.0)));
	else
		tmp = 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, -0.0175], t$95$0, If[LessEqual[y, 0.49], N[(N[(y + x), $MachinePrecision] + N[(z * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.017500000000000002 or 0.48999999999999999 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 69.0

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

   5. Applied rewrites 69.0%

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

   if -0.017500000000000002 < y < 0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0175:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;y \leq 0.49:\\ \;\;\;\;\left(y + x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5e+116) (fma 1.0 z (+ x (sin y))) (* (cos y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5e+116) {
		tmp = fma(1.0, z, (x + sin(y)));
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5e+116)
		tmp = fma(1.0, z, Float64(x + sin(y)));
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5e+116], N[(1.0 * z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.00000000000000025e116

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 90.2%

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

   if 5.00000000000000025e116 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.5% 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 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 69.6

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

5. Applied rewrites 69.6%

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

Reproduce

herbie shell --seed 2024238 
(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))))