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

Percentage Accurate: 99.9% → 99.9%

Time: 7.1s

Alternatives: 12

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 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. Add Preprocessing

Alternative 2: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-37} \lor \neg \left(x \leq 1.5 \cdot 10^{-59}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\cos y \cdot z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.6e-37) (not (<= x 1.5e-59)))
   (fma (/ (* (cos y) z) x) x x)
   (fma (cos y) z (sin y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.6e-37) || !(x <= 1.5e-59)) {
		tmp = fma(((cos(y) * z) / x), x, x);
	} else {
		tmp = fma(cos(y), z, sin(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.6e-37) || !(x <= 1.5e-59))
		tmp = fma(Float64(Float64(cos(y) * z) / x), x, x);
	else
		tmp = fma(cos(y), z, sin(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.6e-37], N[Not[LessEqual[x, 1.5e-59]], $MachinePrecision]], N[(N[(N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z + N[Sin[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6000000000000002e-37 or 1.5e-59 < 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 x around -inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-neg-out N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{z \cdot \cos y}{x}, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 97.6%

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

   if -5.6000000000000002e-37 < x < 1.5e-59

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-37} \lor \neg \left(x \leq 1.5 \cdot 10^{-59}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\cos y \cdot z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -24:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \sin y\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+133}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -24.0)
     t_0
     (if (<= z 8e-57) (fma 1.0 x (sin y)) (if (<= z 3.5e+133) (+ z x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -24.0) {
		tmp = t_0;
	} else if (z <= 8e-57) {
		tmp = fma(1.0, x, sin(y));
	} else if (z <= 3.5e+133) {
		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 <= -24.0)
		tmp = t_0;
	elseif (z <= 8e-57)
		tmp = fma(1.0, x, sin(y));
	elseif (z <= 3.5e+133)
		tmp = Float64(z + 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, -24.0], t$95$0, If[LessEqual[z, 8e-57], N[(1.0 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e+133], N[(z + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -24 or 3.4999999999999998e133 < z

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-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 x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. rgt-mult-inverse N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-cos.f64 N/A

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

   7. Applied rewrites 68.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 84.4%

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

   if -24 < z < 7.99999999999999964e-57

   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 x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. rgt-mult-inverse N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-cos.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 94.3%

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

   if 7.99999999999999964e-57 < z < 3.4999999999999998e133

   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 84.3

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

   5. Applied rewrites 84.3%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -24:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \sin y\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+133}:\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+56} \lor \neg \left(z \leq 3.5 \cdot 10^{+133}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.7e+56) (not (<= z 3.5e+133)))
   (* (cos y) z)
   (fma 1.0 z (+ (sin y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.7e+56) || !(z <= 3.5e+133)) {
		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 <= -3.7e+56) || !(z <= 3.5e+133))
		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, -3.7e+56], N[Not[LessEqual[z, 3.5e+133]], $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 < -3.69999999999999997e56 or 3.4999999999999998e133 < 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. +-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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. rgt-mult-inverse N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-cos.f64 N/A

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 86.5%

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

   if -3.69999999999999997e56 < z < 3.4999999999999998e133

   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 96.3%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+56} \lor \neg \left(z \leq 3.5 \cdot 10^{+133}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+56} \lor \neg \left(z \leq 3.5 \cdot 10^{+133}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.7e+56) (not (<= z 3.5e+133))) (* (cos y) z) (+ z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.7e+56) || !(z <= 3.5e+133)) {
		tmp = cos(y) * z;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.7d+56)) .or. (.not. (z <= 3.5d+133))) then
        tmp = cos(y) * z
    else
        tmp = z + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.7e+56) || !(z <= 3.5e+133)) {
		tmp = Math.cos(y) * z;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.7e+56) or not (z <= 3.5e+133):
		tmp = math.cos(y) * z
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.7e+56) || !(z <= 3.5e+133))
		tmp = Float64(cos(y) * z);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.7e+56) || ~((z <= 3.5e+133)))
		tmp = cos(y) * z;
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.7e+56], N[Not[LessEqual[z, 3.5e+133]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.69999999999999997e56 or 3.4999999999999998e133 < 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. +-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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. rgt-mult-inverse N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. associate-/l* N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-cos.f64 N/A

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 86.5%

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

   if -3.69999999999999997e56 < z < 3.4999999999999998e133

   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 74.9

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

   5. Applied rewrites 74.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+56} \lor \neg \left(z \leq 3.5 \cdot 10^{+133}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+33} \lor \neg \left(y \leq 29000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right), z, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e+33) (not (<= y 29000.0)))
   (+ z x)
   (fma
    (fma
     (fma
      (fma (* y y) -0.001388888888888889 0.041666666666666664)
      (* y y)
      -0.5)
     (* y y)
     1.0)
    z
    (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e+33) || !(y <= 29000.0)) {
		tmp = z + x;
	} else {
		tmp = fma(fma(fma(fma((y * y), -0.001388888888888889, 0.041666666666666664), (y * y), -0.5), (y * y), 1.0), z, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e+33) || !(y <= 29000.0))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(fma(fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664), Float64(y * y), -0.5), Float64(y * y), 1.0), z, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.25e+33], N[Not[LessEqual[y, 29000.0]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.24999999999999993e33 or 29000 < 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 43.5

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

   5. Applied rewrites 43.5%

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

   if -1.24999999999999993e33 < y < 29000

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 96.2

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

   8. Applied rewrites 96.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 96.2

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

   10. Applied rewrites 96.2%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+33} \lor \neg \left(y \leq 29000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right), z, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+33} \lor \neg \left(y \leq 29000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e+33) (not (<= y 29000.0)))
   (+ 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 <= -1.25e+33) || !(y <= 29000.0)) {
		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 <= -1.25e+33) || !(y <= 29000.0))
		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, -1.25e+33], N[Not[LessEqual[y, 29000.0]], $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 < -1.24999999999999993e33 or 29000 < 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 43.5

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

   5. Applied rewrites 43.5%

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

   if -1.24999999999999993e33 < y < 29000

   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 96.0

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

   5. Applied rewrites 96.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+33} \lor \neg \left(y \leq 29000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+34} \lor \neg \left(y \leq 29000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.6e+34) (not (<= y 29000.0)))
   (+ 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 <= -1.6e+34) || !(y <= 29000.0)) {
		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 <= -1.6e+34) || !(y <= 29000.0))
		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, -1.6e+34], N[Not[LessEqual[y, 29000.0]], $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 < -1.5999999999999999e34 or 29000 < 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 43.5

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

   5. Applied rewrites 43.5%

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

   if -1.5999999999999999e34 < y < 29000

   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 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+34} \lor \neg \left(y \leq 29000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -110000000000 \lor \neg \left(y \leq 5000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -110000000000.0) (not (<= y 5000.0)))
   (+ z x)
   (fma (fma (* -0.16666666666666666 y) y 1.0) y (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -110000000000.0) || !(y <= 5000.0)) {
		tmp = z + x;
	} else {
		tmp = fma(fma((-0.16666666666666666 * y), y, 1.0), y, (z + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -110000000000.0) || !(y <= 5000.0))
		tmp = Float64(z + x);
	else
		tmp = fma(fma(Float64(-0.16666666666666666 * y), y, 1.0), y, Float64(z + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -110000000000.0], N[Not[LessEqual[y, 5000.0]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e11 or 5e3 < 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 42.9

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

   5. Applied rewrites 42.9%

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

   if -1.1e11 < y < 5e3

   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 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 97.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -110000000000 \lor \neg \left(y \leq 5000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.8% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+32} \lor \neg \left(y \leq 1.05 \cdot 10^{+28}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9.2e+32) (not (<= y 1.05e+28))) (+ z x) (+ (+ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.2e+32) || !(y <= 1.05e+28)) {
		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 <= (-9.2d+32)) .or. (.not. (y <= 1.05d+28))) 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 <= -9.2e+32) || !(y <= 1.05e+28)) {
		tmp = z + x;
	} else {
		tmp = (y + x) + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -9.2e+32) or not (y <= 1.05e+28):
		tmp = z + x
	else:
		tmp = (y + x) + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -9.2e+32) || !(y <= 1.05e+28))
		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 <= -9.2e+32) || ~((y <= 1.05e+28)))
		tmp = z + x;
	else
		tmp = (y + x) + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -9.2e+32], N[Not[LessEqual[y, 1.05e+28]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.1999999999999998e32 or 1.04999999999999995e28 < 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 42.5

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

   5. Applied rewrites 42.5%

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

   if -9.1999999999999998e32 < y < 1.04999999999999995e28

   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 93.6

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

   5. Applied rewrites 93.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+32} \lor \neg \left(y \leq 1.05 \cdot 10^{+28}\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.6% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 67.2

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

5. Applied rewrites 67.2%

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

6. Final simplification 67.2%

   \[\leadsto z + x \]
7. Add Preprocessing

Alternative 12: 29.5% 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 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 56.1

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 27.6%

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

9. Final simplification 27.6%

   \[\leadsto z + y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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))))