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

Percentage Accurate: 99.9% → 99.9%

Time: 7.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision] + 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. Final simplification 99.9%

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

Alternative 2: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -14500000:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y + \cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -14500000.0)
   (fma (cos y) z (+ y x))
   (if (<= z 7.8e-19) (fma 1.0 z (+ (sin y) x)) (+ (sin y) (* (cos y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -14500000.0) {
		tmp = fma(cos(y), z, (y + x));
	} else if (z <= 7.8e-19) {
		tmp = fma(1.0, z, (sin(y) + x));
	} else {
		tmp = sin(y) + (cos(y) * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -14500000.0)
		tmp = fma(cos(y), z, Float64(y + x));
	elseif (z <= 7.8e-19)
		tmp = fma(1.0, z, Float64(sin(y) + x));
	else
		tmp = Float64(sin(y) + Float64(cos(y) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -14500000.0], N[(N[Cos[y], $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.8e-19], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.45e7

   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 85.0

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

   5. Applied rewrites 85.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if -1.45e7 < z < 7.7999999999999999e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 7.7999999999999999e-19 < z

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   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. lower-sin.f64 93.1

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

   5. Applied rewrites 93.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -14500000:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y + \cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -14500000:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -14500000.0)
   (fma (cos y) z (+ y x))
   (if (<= z 7.8e-19) (fma 1.0 z (+ (sin y) x)) (fma (cos y) z (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -14500000.0) {
		tmp = fma(cos(y), z, (y + x));
	} else if (z <= 7.8e-19) {
		tmp = fma(1.0, z, (sin(y) + x));
	} else {
		tmp = fma(cos(y), z, sin(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -14500000.0)
		tmp = fma(cos(y), z, Float64(y + x));
	elseif (z <= 7.8e-19)
		tmp = fma(1.0, z, Float64(sin(y) + x));
	else
		tmp = fma(cos(y), z, sin(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -14500000.0], N[(N[Cos[y], $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.8e-19], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z + N[Sin[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.45e7

   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 85.0

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

   5. Applied rewrites 85.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if -1.45e7 < z < 7.7999999999999999e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 7.7999999999999999e-19 < z

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   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 93.1

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

   5. Applied rewrites 93.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -14500000:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

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

Wolfram

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

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 5: 76.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-23}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-262}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.15e-23)
   (+ z x)
   (if (<= x -7.5e-262)
     (* (cos y) z)
     (if (<= x 1.7e-16) (fma 1.0 z (sin y)) (+ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.15e-23) {
		tmp = z + x;
	} else if (x <= -7.5e-262) {
		tmp = cos(y) * z;
	} else if (x <= 1.7e-16) {
		tmp = fma(1.0, z, sin(y));
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.15e-23)
		tmp = Float64(z + x);
	elseif (x <= -7.5e-262)
		tmp = Float64(cos(y) * z);
	elseif (x <= 1.7e-16)
		tmp = fma(1.0, z, sin(y));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.15e-23], N[(z + x), $MachinePrecision], If[LessEqual[x, -7.5e-262], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], If[LessEqual[x, 1.7e-16], N[(1.0 * z + N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.15000000000000001e-23 or 1.7e-16 < 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 89.8

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

   5. Applied rewrites 89.8%

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

   if -2.15000000000000001e-23 < x < -7.5000000000000002e-262

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. inv-pow N/A

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

      13. lower-pow.f64 N/A

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

   6. Applied rewrites 65.6%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 67.5%

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

   if -7.5000000000000002e-262 < x < 1.7e-16

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 82.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 82.0

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

   7. Applied rewrites 82.0%

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

   8. Taylor expanded in x around 0

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

      1. lower-sin.f64 79.6

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

   10. Applied rewrites 79.6%

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

Alternative 6: 88.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -14500000:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -14500000.0)
   (fma (cos y) z (+ y x))
   (if (<= z 5.3e+42) (fma 1.0 z (+ (sin y) x)) (* (cos y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -14500000.0) {
		tmp = fma(cos(y), z, (y + x));
	} else if (z <= 5.3e+42) {
		tmp = fma(1.0, z, (sin(y) + x));
	} else {
		tmp = cos(y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -14500000.0)
		tmp = fma(cos(y), z, Float64(y + x));
	elseif (z <= 5.3e+42)
		tmp = fma(1.0, z, Float64(sin(y) + x));
	else
		tmp = Float64(cos(y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -14500000.0], N[(N[Cos[y], $MachinePrecision] * z + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.3e+42], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.45e7

   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 85.0

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

   5. Applied rewrites 85.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if -1.45e7 < z < 5.30000000000000028e42

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.0

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

   7. Applied rewrites 97.0%

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

   if 5.30000000000000028e42 < z

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. inv-pow N/A

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

      13. lower-pow.f64 N/A

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

   6. Applied rewrites 41.0%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 95.0%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -14500000:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, y + x\right)\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -7 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -7e+92) t_0 (if (<= z 5.3e+42) (fma 1.0 z (+ (sin y) x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -7e+92) {
		tmp = t_0;
	} else if (z <= 5.3e+42) {
		tmp = fma(1.0, z, (sin(y) + x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -7e+92)
		tmp = t_0;
	elseif (z <= 5.3e+42)
		tmp = fma(1.0, z, Float64(sin(y) + 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, -7e+92], t$95$0, If[LessEqual[z, 5.3e+42], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.99999999999999972e92 or 5.30000000000000028e42 < z

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. inv-pow N/A

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

      13. lower-pow.f64 N/A

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

   6. Applied rewrites 37.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 90.1%

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

   if -6.99999999999999972e92 < z < 5.30000000000000028e42

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 94.8

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

   7. Applied rewrites 94.8%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+92}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -7 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-19}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -7e+92) t_0 (if (<= z 7.8e-19) (+ z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -7e+92) {
		tmp = t_0;
	} else if (z <= 7.8e-19) {
		tmp = z + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(y) * z
    if (z <= (-7d+92)) then
        tmp = t_0
    else if (z <= 7.8d-19) then
        tmp = z + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * z;
	double tmp;
	if (z <= -7e+92) {
		tmp = t_0;
	} else if (z <= 7.8e-19) {
		tmp = z + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * z
	tmp = 0
	if z <= -7e+92:
		tmp = t_0
	elif z <= 7.8e-19:
		tmp = z + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -7e+92)
		tmp = t_0;
	elseif (z <= 7.8e-19)
		tmp = Float64(z + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * z;
	tmp = 0.0;
	if (z <= -7e+92)
		tmp = t_0;
	elseif (z <= 7.8e-19)
		tmp = z + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -7e+92], t$95$0, If[LessEqual[z, 7.8e-19], N[(z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.99999999999999972e92 or 7.7999999999999999e-19 < z

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. inv-pow N/A

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

      13. lower-pow.f64 N/A

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

   6. Applied rewrites 43.0%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 86.2%

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

   if -6.99999999999999972e92 < z < 7.7999999999999999e-19

   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 73.6

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

   5. Applied rewrites 73.6%

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

Alternative 9: 70.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 7500:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, z + x\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -11.0)
   (+ z x)
   (if (<= y 7500.0) (fma (fma (* -0.5 y) z 1.0) y (+ z x)) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -11.0) {
		tmp = z + x;
	} else if (y <= 7500.0) {
		tmp = fma(fma((-0.5 * y), z, 1.0), y, (z + x));
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -11.0)
		tmp = Float64(z + x);
	elseif (y <= 7500.0)
		tmp = fma(fma(Float64(-0.5 * y), z, 1.0), y, Float64(z + x));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -11.0], N[(z + x), $MachinePrecision], If[LessEqual[y, 7500.0], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -11 or 7500 < 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 42.8

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

   5. Applied rewrites 42.8%

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

   if -11 < y < 7500

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 10: 69.9% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+61}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+65}:\\ \;\;\;\;\left(y + x\right) + z\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.05e+61) (+ z x) (if (<= y 4.6e+65) (+ (+ y x) z) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.05e+61) {
		tmp = z + x;
	} else if (y <= 4.6e+65) {
		tmp = (y + x) + 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 (y <= (-1.05d+61)) then
        tmp = z + x
    else if (y <= 4.6d+65) then
        tmp = (y + x) + 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 (y <= -1.05e+61) {
		tmp = z + x;
	} else if (y <= 4.6e+65) {
		tmp = (y + x) + z;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.05e+61:
		tmp = z + x
	elif y <= 4.6e+65:
		tmp = (y + x) + z
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.05e+61)
		tmp = Float64(z + x);
	elseif (y <= 4.6e+65)
		tmp = Float64(Float64(y + x) + z);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.05e+61)
		tmp = z + x;
	elseif (y <= 4.6e+65)
		tmp = (y + x) + z;
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.05e+61], N[(z + x), $MachinePrecision], If[LessEqual[y, 4.6e+65], N[(N[(y + x), $MachinePrecision] + z), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0500000000000001e61 or 4.6e65 < 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 43.2

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

   5. Applied rewrites 43.2%

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

   if -1.0500000000000001e61 < y < 4.6e65

   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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 86.5

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

   5. Applied rewrites 86.5%

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

Alternative 11: 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 68.3

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

5. Applied rewrites 68.3%

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

Reproduce

herbie shell --seed 2024296 
(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))))