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

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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. Final simplification 99.9%

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

Alternative 2: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y + \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;z \leq -680000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z (cos y)) (/ 1.0 (/ 1.0 x)))))
   (if (<= z -680000000000.0)
     t_0
     (if (<= z 0.0005) (fma 1.0 z (+ x (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * cos(y)) + (1.0 / (1.0 / x));
	double tmp;
	if (z <= -680000000000.0) {
		tmp = t_0;
	} else if (z <= 0.0005) {
		tmp = fma(1.0, z, (x + sin(y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * cos(y)) + Float64(1.0 / Float64(1.0 / x)))
	tmp = 0.0
	if (z <= -680000000000.0)
		tmp = t_0;
	elseif (z <= 0.0005)
		tmp = fma(1.0, z, Float64(x + sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -680000000000.0], t$95$0, If[LessEqual[z, 0.0005], N[(1.0 * z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8e11 or 5.0000000000000001e-4 < z

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

         \[\leadsto \frac{1}{\frac{\sin y - x}{\color{blue}{{\sin y}^{2}} - x \cdot x}} + z \cdot \cos y \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sin y - x}{\color{blue}{{\sin y}^{2}} - x \cdot x}} + z \cdot \cos y \]

      11. lower-*.f64 76.5

          \[\leadsto \frac{1}{\frac{\sin y - x}{{\sin y}^{2} - \color{blue}{x \cdot x}}} + z \cdot \cos y \]

   4. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin y - x}{{\sin y}^{2} - x \cdot x}}} + z \cdot \cos y \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if -6.8e11 < z < 5.0000000000000001e-4

   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 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -680000000000:\\ \;\;\;\;z \cdot \cos y + \frac{1}{\frac{1}{x}}\\ \mathbf{elif}\;z \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y + \frac{1}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 230000:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\cos y}{x} \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (cos y) z (+ x y))))
   (if (<= z -3.6e+83)
     t_0
     (if (<= z 230000.0)
       (fma 1.0 z (+ x (sin y)))
       (if (<= z 9e+108) (fma (* (/ (cos y) x) z) x x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fma(cos(y), z, (x + y));
	double tmp;
	if (z <= -3.6e+83) {
		tmp = t_0;
	} else if (z <= 230000.0) {
		tmp = fma(1.0, z, (x + sin(y)));
	} else if (z <= 9e+108) {
		tmp = fma(((cos(y) / x) * z), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(cos(y), z, Float64(x + y))
	tmp = 0.0
	if (z <= -3.6e+83)
		tmp = t_0;
	elseif (z <= 230000.0)
		tmp = fma(1.0, z, Float64(x + sin(y)));
	elseif (z <= 9e+108)
		tmp = fma(Float64(Float64(cos(y) / x) * z), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e+83], t$95$0, If[LessEqual[z, 230000.0], N[(1.0 * z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e+108], N[(N[(N[(N[Cos[y], $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5999999999999997e83 or 9e108 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 87.1

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

   5. Applied rewrites 87.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 87.1

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

   7. Applied rewrites 87.1%

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

   if -3.5999999999999997e83 < z < 2.3e5

   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 95.8%

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

   if 2.3e5 < z < 9e108

   1. Initial program 99.5%

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

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

         \[\leadsto \frac{1}{\frac{\sin y - x}{\color{blue}{{\sin y}^{2}} - x \cdot x}} + z \cdot \cos y \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sin y - x}{\color{blue}{{\sin y}^{2}} - x \cdot x}} + z \cdot \cos y \]

      11. lower-*.f64 91.4

          \[\leadsto \frac{1}{\frac{\sin y - x}{{\sin y}^{2} - \color{blue}{x \cdot x}}} + z \cdot \cos y \]

   4. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin y - x}{{\sin y}^{2} - x \cdot x}}} + z \cdot \cos y \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 95.3

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

   10. Applied rewrites 95.3%

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

   11. Taylor expanded in z around inf

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

   13. Applied rewrites 95.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{elif}\;z \leq 230000:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\cos y}{x} \cdot z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (cos y) z (+ x y))))
   (if (<= z -3.6e+83)
     t_0
     (if (<= z 1.45e+70) (fma 1.0 z (+ x (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(cos(y), z, (x + y));
	double tmp;
	if (z <= -3.6e+83) {
		tmp = t_0;
	} else if (z <= 1.45e+70) {
		tmp = fma(1.0, z, (x + sin(y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(cos(y), z, Float64(x + y))
	tmp = 0.0
	if (z <= -3.6e+83)
		tmp = t_0;
	elseif (z <= 1.45e+70)
		tmp = fma(1.0, z, Float64(x + sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e+83], t$95$0, If[LessEqual[z, 1.45e+70], N[(1.0 * z + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5999999999999997e83 or 1.4499999999999999e70 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 84.5

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

   5. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\left(y + x\right)} + z \cdot \cos y \]
   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 84.5

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

   7. Applied rewrites 84.5%

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

   if -3.5999999999999997e83 < z < 1.4499999999999999e70

   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 y around 0

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

   7. Applied rewrites 93.8%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(1, z, x + \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sin y\\ \mathbf{if}\;y \leq -0.0135:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 86000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (sin y))))
   (if (<= y -0.0135)
     t_0
     (if (<= y 86000000000.0)
       (fma (fma (* -0.16666666666666666 y) y 1.0) y (+ x z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + sin(y);
	double tmp;
	if (y <= -0.0135) {
		tmp = t_0;
	} else if (y <= 86000000000.0) {
		tmp = fma(fma((-0.16666666666666666 * y), y, 1.0), y, (x + z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + sin(y))
	tmp = 0.0
	if (y <= -0.0135)
		tmp = t_0;
	elseif (y <= 86000000000.0)
		tmp = fma(fma(Float64(-0.16666666666666666 * y), y, 1.0), y, Float64(x + z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0135], t$95$0, If[LessEqual[y, 86000000000.0], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0134999999999999998 or 8.6e10 < y

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

         \[\leadsto \frac{1}{\frac{\sin y - x}{\color{blue}{{\sin y}^{2}} - x \cdot x}} + z \cdot \cos y \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sin y - x}{\color{blue}{{\sin y}^{2}} - x \cdot x}} + z \cdot \cos y \]

      11. lower-*.f64 75.8

          \[\leadsto \frac{1}{\frac{\sin y - x}{{\sin y}^{2} - \color{blue}{x \cdot x}}} + z \cdot \cos y \]

   4. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin y - x}{{\sin y}^{2} - x \cdot x}}} + z \cdot \cos y \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 76.1

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

   7. Applied rewrites 76.1%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 83.4

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

   10. Applied rewrites 83.4%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 55.5%

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

   if -0.0134999999999999998 < y < 8.6e10

   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

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

         \[\leadsto \frac{1}{\frac{\sin y - x}{\color{blue}{{\sin y}^{2}} - x \cdot x}} + z \cdot \cos y \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sin y - x}{\color{blue}{{\sin y}^{2}} - x \cdot x}} + z \cdot \cos y \]

      11. lower-*.f64 73.3

          \[\leadsto \frac{1}{\frac{\sin y - x}{{\sin y}^{2} - \color{blue}{x \cdot x}}} + z \cdot \cos y \]

   4. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin y - x}{{\sin y}^{2} - x \cdot x}}} + z \cdot \cos y \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 92.3

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in y around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 97.9

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

   10. Applied rewrites 97.9%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 98.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0135:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;y \leq 86000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, z, x + \sin y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma 1.0 z (+ x (sin y))))

C

double code(double x, double y, double z) {
	return fma(1.0, z, (x + sin(y)));
}

Julia

function code(x, y, z)
	return fma(1.0, z, Float64(x + sin(y)))
end

Wolfram

code[x_, y_, z_] := N[(1.0 * z + N[(x + N[Sin[y], $MachinePrecision]), $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. 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 80.2%

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

8. Final simplification 80.2%

   \[\leadsto \mathsf{fma}\left(1, z, x + \sin y\right) \]
9. Add Preprocessing

Alternative 7: 70.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 4400000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -17.0)
   (+ x z)
   (if (<= y 4400000.0)
     (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ x z))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -17.0) {
		tmp = x + z;
	} else if (y <= 4400000.0) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -17.0)
		tmp = Float64(x + z);
	elseif (y <= 4400000.0)
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -17.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 4400000.0], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -17 or 4.4e6 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 37.1

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

   5. Applied rewrites 37.1%

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

   if -17 < y < 4.4e6

   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 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 4400000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+48}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 15500000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3e+48)
   (+ x z)
   (if (<= y 15500000000000.0)
     (fma (fma (* -0.16666666666666666 y) y 1.0) y (+ x z))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e+48) {
		tmp = x + z;
	} else if (y <= 15500000000000.0) {
		tmp = fma(fma((-0.16666666666666666 * y), y, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3e+48)
		tmp = Float64(x + z);
	elseif (y <= 15500000000000.0)
		tmp = fma(fma(Float64(-0.16666666666666666 * y), y, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3e+48], N[(x + z), $MachinePrecision], If[LessEqual[y, 15500000000000.0], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3e48 or 1.55e13 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 38.4

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

   5. Applied rewrites 38.4%

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

   if -3e48 < y < 1.55e13

   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

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

         \[\leadsto \frac{1}{\frac{\sin y - x}{\color{blue}{{\sin y}^{2}} - x \cdot x}} + z \cdot \cos y \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sin y - x}{\color{blue}{{\sin y}^{2}} - x \cdot x}} + z \cdot \cos y \]

      11. lower-*.f64 74.2

          \[\leadsto \frac{1}{\frac{\sin y - x}{{\sin y}^{2} - \color{blue}{x \cdot x}}} + z \cdot \cos y \]

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin y - x}{{\sin y}^{2} - x \cdot x}}} + z \cdot \cos y \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 91.1

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in y around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 92.2

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

   10. Applied rewrites 92.2%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 92.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+48}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 15500000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 4500000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -17.0)
   (+ x z)
   (if (<= y 4500000.0) (fma (fma (* -0.5 y) z 1.0) y (+ x z)) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -17.0) {
		tmp = x + z;
	} else if (y <= 4500000.0) {
		tmp = fma(fma((-0.5 * y), z, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -17.0)
		tmp = Float64(x + z);
	elseif (y <= 4500000.0)
		tmp = fma(fma(Float64(-0.5 * y), z, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -17.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 4500000.0], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -17 or 4.5e6 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 37.1

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

   5. Applied rewrites 37.1%

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

   if -17 < y < 4.5e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 4500000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.7% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+17}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-35}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.6e+17) (+ x z) (if (<= y 2.5e-35) (+ (+ x y) z) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.6e+17) {
		tmp = x + z;
	} else if (y <= 2.5e-35) {
		tmp = (x + y) + z;
	} else {
		tmp = 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 <= (-8.6d+17)) then
        tmp = x + z
    else if (y <= 2.5d-35) then
        tmp = (x + y) + z
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.6e+17) {
		tmp = x + z;
	} else if (y <= 2.5e-35) {
		tmp = (x + y) + z;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.6e+17:
		tmp = x + z
	elif y <= 2.5e-35:
		tmp = (x + y) + z
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.6e+17)
		tmp = Float64(x + z);
	elseif (y <= 2.5e-35)
		tmp = Float64(Float64(x + y) + z);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.6e+17)
		tmp = x + z;
	elseif (y <= 2.5e-35)
		tmp = (x + y) + z;
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.6e+17], N[(x + z), $MachinePrecision], If[LessEqual[y, 2.5e-35], N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.6e17 or 2.49999999999999982e-35 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 42.5

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

   5. Applied rewrites 42.5%

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

   if -8.6e17 < y < 2.49999999999999982e-35

   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 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+17}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-35}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.3% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

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 + z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + z), $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 65.4

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

5. Applied rewrites 65.4%

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

6. Final simplification 65.4%

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

Reproduce

herbie shell --seed 2024294 
(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))))