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

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 81.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -5000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (<= t_0 -5000.0)
     (+ z x)
     (if (<= t_0 -0.05)
       (sin y)
       (if (<= t_0 1e-5)
         (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) 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)) + (z * cos(y));
	double tmp;
	if (t_0 <= -5000.0) {
		tmp = z + x;
	} else if (t_0 <= -0.05) {
		tmp = sin(y);
	} else if (t_0 <= 1e-5) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (z + x));
	} else if (t_0 <= 1.0) {
		tmp = sin(y);
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	tmp = 0.0
	if (t_0 <= -5000.0)
		tmp = Float64(z + x);
	elseif (t_0 <= -0.05)
		tmp = sin(y);
	elseif (t_0 <= 1e-5)
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(z + x));
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5000.0], N[(z + x), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 1e-5], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * 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))) < -5e3 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -5e3 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.050000000000000003 or 1.00000000000000008e-5 < (+.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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 94.6%

       \[\leadsto \sin y \]

   if -0.050000000000000003 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1.00000000000000008e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 3: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-7} \lor \neg \left(x \leq 4.6 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.35e-7) (not (<= x 4.6e-7)))
   (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 ((x <= -1.35e-7) || !(x <= 4.6e-7)) {
		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 ((x <= -1.35e-7) || !(x <= 4.6e-7))
		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[Or[LessEqual[x, -1.35e-7], N[Not[LessEqual[x, 4.6e-7]], $MachinePrecision]], 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 2 regimes

2. if x < -1.35000000000000004e-7 or 4.5999999999999999e-7 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 90.7%

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

   if -1.35000000000000004e-7 < x < 4.5999999999999999e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 93.0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-7} \lor \neg \left(x \leq 4.6 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -1.26 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-10}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-71}:\\ \;\;\;\;\sin y + x\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+30}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -1.26e+48)
     t_0
     (if (<= z -1.8e-10)
       (+ z x)
       (if (<= z 8.6e-71) (+ (sin y) x) (if (<= z 1.65e+30) (+ z x) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -1.26e+48) {
		tmp = t_0;
	} else if (z <= -1.8e-10) {
		tmp = z + x;
	} else if (z <= 8.6e-71) {
		tmp = sin(y) + x;
	} else if (z <= 1.65e+30) {
		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 <= (-1.26d+48)) then
        tmp = t_0
    else if (z <= (-1.8d-10)) then
        tmp = z + x
    else if (z <= 8.6d-71) then
        tmp = sin(y) + x
    else if (z <= 1.65d+30) 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 <= -1.26e+48) {
		tmp = t_0;
	} else if (z <= -1.8e-10) {
		tmp = z + x;
	} else if (z <= 8.6e-71) {
		tmp = Math.sin(y) + x;
	} else if (z <= 1.65e+30) {
		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 <= -1.26e+48:
		tmp = t_0
	elif z <= -1.8e-10:
		tmp = z + x
	elif z <= 8.6e-71:
		tmp = math.sin(y) + x
	elif z <= 1.65e+30:
		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 <= -1.26e+48)
		tmp = t_0;
	elseif (z <= -1.8e-10)
		tmp = Float64(z + x);
	elseif (z <= 8.6e-71)
		tmp = Float64(sin(y) + x);
	elseif (z <= 1.65e+30)
		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 <= -1.26e+48)
		tmp = t_0;
	elseif (z <= -1.8e-10)
		tmp = z + x;
	elseif (z <= 8.6e-71)
		tmp = sin(y) + x;
	elseif (z <= 1.65e+30)
		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, -1.26e+48], t$95$0, If[LessEqual[z, -1.8e-10], N[(z + x), $MachinePrecision], If[LessEqual[z, 8.6e-71], N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.65e+30], N[(z + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.26000000000000001e48 or 1.65000000000000013e30 < 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

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 81.0

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

   7. Applied rewrites 81.0%

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

   if -1.26000000000000001e48 < z < -1.8e-10 or 8.5999999999999994e-71 < z < 1.65000000000000013e30

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 86.5

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

   5. Applied rewrites 86.5%

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

   if -1.8e-10 < z < 8.5999999999999994e-71

   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

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 94.6

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

   7. Applied rewrites 94.6%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+48}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-10}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-71}:\\ \;\;\;\;\sin y + x\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+30}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+48} \lor \neg \left(z \leq 1.65 \cdot 10^{+30}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.26e+48) (not (<= z 1.65e+30)))
   (* (cos y) z)
   (fma 1.0 z (+ (sin y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.26e+48) || !(z <= 1.65e+30)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(1.0, z, (sin(y) + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.26e+48) || !(z <= 1.65e+30))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(1.0, z, Float64(sin(y) + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.26e+48], N[Not[LessEqual[z, 1.65e+30]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.26000000000000001e48 or 1.65000000000000013e30 < 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

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 81.0

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

   7. Applied rewrites 81.0%

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

   if -1.26000000000000001e48 < z < 1.65000000000000013e30

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 97.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+48} \lor \neg \left(z \leq 1.65 \cdot 10^{+30}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+48}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.26e+48)
   (* (cos y) z)
   (if (<= z 1.26e+39) (fma 1.0 z (+ (sin y) x)) (fma (cos y) z (+ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.26e+48) {
		tmp = cos(y) * z;
	} else if (z <= 1.26e+39) {
		tmp = fma(1.0, z, (sin(y) + x));
	} else {
		tmp = fma(cos(y), z, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.26e+48)
		tmp = Float64(cos(y) * z);
	elseif (z <= 1.26e+39)
		tmp = fma(1.0, z, Float64(sin(y) + x));
	else
		tmp = fma(cos(y), z, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.26e+48], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 1.26e+39], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.26000000000000001e48

   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

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 85.3

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

   7. Applied rewrites 85.3%

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

   if -1.26000000000000001e48 < z < 1.26000000000000001e39

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 96.1%

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

   if 1.26000000000000001e39 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 82.3

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

   5. Applied rewrites 82.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 82.3

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

   7. Applied rewrites 82.3%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+48}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -380000000000 \lor \neg \left(y \leq 3.5 \cdot 10^{+17}\right):\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -380000000000.0) (not (<= y 3.5e+17)))
   (+ (sin y) x)
   (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -380000000000.0) || !(y <= 3.5e+17)) {
		tmp = sin(y) + x;
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -380000000000.0) || !(y <= 3.5e+17))
		tmp = Float64(sin(y) + x);
	else
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -380000000000.0], N[Not[LessEqual[y, 3.5e+17]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8e11 or 3.5e17 < y

   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

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \sin y} \]
   6. 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 63.3

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

   7. Applied rewrites 63.3%

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

   if -3.8e11 < y < 3.5e17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -380000000000 \lor \neg \left(y \leq 3.5 \cdot 10^{+17}\right):\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -84000000000000 \lor \neg \left(y \leq 3.2 \cdot 10^{+21}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -84000000000000.0) (not (<= y 3.2e+21)))
   (+ z x)
   (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -84000000000000.0) || !(y <= 3.2e+21)) {
		tmp = z + x;
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -84000000000000.0) || !(y <= 3.2e+21))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -84000000000000.0], N[Not[LessEqual[y, 3.2e+21]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.4e13 or 3.2e21 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 37.3

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

   5. Applied rewrites 37.3%

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

   if -8.4e13 < y < 3.2e21

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.2

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

   5. Applied rewrites 96.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -84000000000000 \lor \neg \left(y \leq 3.2 \cdot 10^{+21}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+22} \lor \neg \left(y \leq 1.05 \cdot 10^{+22}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.05e+22) (not (<= y 1.05e+22)))
   (+ z x)
   (fma (fma (* -0.5 y) z 1.0) y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e+22) || !(y <= 1.05e+22)) {
		tmp = z + x;
	} else {
		tmp = fma(fma((-0.5 * y), z, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.05e+22) || !(y <= 1.05e+22))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(Float64(-0.5 * y), z, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.05e+22], N[Not[LessEqual[y, 1.05e+22]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0499999999999999e22 or 1.0499999999999999e22 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 37.6

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

   5. Applied rewrites 37.6%

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

   if -1.0499999999999999e22 < y < 1.0499999999999999e22

   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 95.3

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

   5. Applied rewrites 95.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+22} \lor \neg \left(y \leq 1.05 \cdot 10^{+22}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.5% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+22} \lor \neg \left(y \leq 3.2\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4e+22) (not (<= y 3.2))) (+ z x) (+ (+ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e+22) || !(y <= 3.2)) {
		tmp = z + x;
	} else {
		tmp = (y + x) + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-4d+22)) .or. (.not. (y <= 3.2d0))) then
        tmp = z + x
    else
        tmp = (y + x) + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4e+22) || !(y <= 3.2)) {
		tmp = z + x;
	} else {
		tmp = (y + x) + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4e+22) or not (y <= 3.2):
		tmp = z + x
	else:
		tmp = (y + x) + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4e+22) || !(y <= 3.2))
		tmp = Float64(z + x);
	else
		tmp = Float64(Float64(y + x) + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4e+22) || ~((y <= 3.2)))
		tmp = z + x;
	else
		tmp = (y + x) + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4e+22], N[Not[LessEqual[y, 3.2]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4e22 or 3.2000000000000002 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 36.9

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

   5. Applied rewrites 36.9%

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

   if -4e22 < y < 3.2000000000000002

   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 96.5

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

   5. Applied rewrites 96.5%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+22} \lor \neg \left(y \leq 3.2\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.4% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -64000 \lor \neg \left(x \leq 3 \cdot 10^{+44}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -64000.0) (not (<= x 3e+44))) (+ x y) (+ z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -64000.0) || !(x <= 3e+44)) {
		tmp = x + y;
	} else {
		tmp = z + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-64000.0d0)) .or. (.not. (x <= 3d+44))) then
        tmp = x + y
    else
        tmp = z + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -64000.0) || !(x <= 3e+44)) {
		tmp = x + y;
	} else {
		tmp = z + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -64000.0) or not (x <= 3e+44):
		tmp = x + y
	else:
		tmp = z + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -64000.0) || !(x <= 3e+44))
		tmp = Float64(x + y);
	else
		tmp = Float64(z + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -64000.0) || ~((x <= 3e+44)))
		tmp = x + y;
	else
		tmp = z + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -64000.0], N[Not[LessEqual[x, 3e+44]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -64000 or 2.99999999999999987e44 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \left(z + y \cdot \left(1 + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right)} \]
   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 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 2.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 71.4%

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

   if -64000 < x < 2.99999999999999987e44

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 38.6%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -64000 \lor \neg \left(x \leq 3 \cdot 10^{+44}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z + y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.3% 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 61.1

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

5. Applied rewrites 61.1%

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

Alternative 13: 29.9% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-sin.f64 61.6

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

5. Applied rewrites 61.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 26.6%

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

Reproduce

herbie shell --seed 2024318 
(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))))