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

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 70.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -0.95 \lor \neg \left(t\_0 \leq 0.02\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (or (<= t_0 -0.95) (not (<= t_0 0.02))) (+ z x) (+ (+ y x) z))))

C

double code(double x, double y, double z) {
	double t_0 = (x + sin(y)) + (z * cos(y));
	double tmp;
	if ((t_0 <= -0.95) || !(t_0 <= 0.02)) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (x + sin(y)) + (z * cos(y))
    if ((t_0 <= (-0.95d0)) .or. (.not. (t_0 <= 0.02d0))) 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 t_0 = (x + Math.sin(y)) + (z * Math.cos(y));
	double tmp;
	if ((t_0 <= -0.95) || !(t_0 <= 0.02)) {
		tmp = z + x;
	} else {
		tmp = (y + x) + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + math.sin(y)) + (z * math.cos(y))
	tmp = 0
	if (t_0 <= -0.95) or not (t_0 <= 0.02):
		tmp = z + x
	else:
		tmp = (y + x) + z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	tmp = 0.0
	if ((t_0 <= -0.95) || !(t_0 <= 0.02))
		tmp = Float64(z + x);
	else
		tmp = Float64(Float64(y + x) + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + sin(y)) + (z * cos(y));
	tmp = 0.0;
	if ((t_0 <= -0.95) || ~((t_0 <= 0.02)))
		tmp = z + x;
	else
		tmp = (y + x) + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.95], N[Not[LessEqual[t$95$0, 0.02]], $MachinePrecision]], N[(z + x), $MachinePrecision], N[(N[(y + x), $MachinePrecision] + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.94999999999999996 or 0.0200000000000000004 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -0.94999999999999996 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 0.0200000000000000004

   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 77.9

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

   5. Applied rewrites 77.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + z \cdot \cos y \leq -0.95 \lor \neg \left(\left(x + \sin y\right) + z \cdot \cos y \leq 0.02\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) + z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e-85)
   (fma 1.0 z (+ (sin y) x))
   (if (<= x 1.1e+50) (fma (cos y) z (sin y)) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e-85) {
		tmp = fma(1.0, z, (sin(y) + x));
	} else if (x <= 1.1e+50) {
		tmp = fma(cos(y), z, sin(y));
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3e-85)
		tmp = fma(1.0, z, Float64(sin(y) + x));
	elseif (x <= 1.1e+50)
		tmp = fma(cos(y), z, sin(y));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e-85], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e+50], N[(N[Cos[y], $MachinePrecision] * z + N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.00000000000000022e-85

   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 94.5%

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

   if -3.00000000000000022e-85 < x < 1.10000000000000008e50

   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 96.6

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

   5. Applied rewrites 96.6%

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

   if 1.10000000000000008e50 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 92.6

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

   5. Applied rewrites 92.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+55} \lor \neg \left(z \leq 1.35 \cdot 10^{+112}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9e+55) (not (<= z 1.35e+112)))
   (fma (cos y) z (+ x y))
   (fma 1.0 z (+ (sin y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9e+55) || !(z <= 1.35e+112)) {
		tmp = fma(cos(y), z, (x + y));
	} else {
		tmp = fma(1.0, z, (sin(y) + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9e+55) || !(z <= 1.35e+112))
		tmp = fma(cos(y), z, Float64(x + y));
	else
		tmp = fma(1.0, z, Float64(sin(y) + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9e+55], N[Not[LessEqual[z, 1.35e+112]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999996e55 or 1.3500000000000001e112 < 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 91.6

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

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\left(y + x\right)} + z \cdot \cos y \]
   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 91.6

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

   7. Applied rewrites 91.6%

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

   if -8.99999999999999996e55 < z < 1.3500000000000001e112

   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 93.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+55} \lor \neg \left(z \leq 1.35 \cdot 10^{+112}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+179} \lor \neg \left(z \leq 5.2 \cdot 10^{+179}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e+179) (not (<= z 5.2e+179)))
   (* (cos y) z)
   (fma 1.0 z (+ (sin y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e+179) || !(z <= 5.2e+179)) {
		tmp = cos(y) * z;
	} else {
		tmp = fma(1.0, z, (sin(y) + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.25e+179) || !(z <= 5.2e+179))
		tmp = Float64(cos(y) * z);
	else
		tmp = fma(1.0, z, Float64(sin(y) + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.25e+179], N[Not[LessEqual[z, 5.2e+179]], $MachinePrecision]], N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision], N[(1.0 * z + N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e179 or 5.2000000000000004e179 < 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. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 91.9

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

   7. Applied rewrites 91.9%

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

   if -1.25e179 < z < 5.2000000000000004e179

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 90.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+179} \lor \neg \left(z \leq 5.2 \cdot 10^{+179}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, z, \sin y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -27500000000000 \lor \neg \left(y \leq 0.4\right):\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -27500000000000.0) (not (<= y 0.4)))
   (+ (sin y) 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 <= -27500000000000.0) || !(y <= 0.4)) {
		tmp = sin(y) + 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 <= -27500000000000.0) || !(y <= 0.4))
		tmp = Float64(sin(y) + 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, -27500000000000.0], N[Not[LessEqual[y, 0.4]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] + 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 < -2.75e13 or 0.40000000000000002 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-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 92.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 70.6%

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

   if -2.75e13 < y < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -27500000000000 \lor \neg \left(y \leq 0.4\right):\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -38000000000000 \lor \neg \left(y \leq 155000000000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -38000000000000.0) (not (<= y 155000000000.0)))
   (+ 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 <= -38000000000000.0) || !(y <= 155000000000.0)) {
		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 <= -38000000000000.0) || !(y <= 155000000000.0))
		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, -38000000000000.0], N[Not[LessEqual[y, 155000000000.0]], $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 < -3.8e13 or 1.55e11 < 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 47.8

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

   5. Applied rewrites 47.8%

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

   if -3.8e13 < y < 1.55e11

   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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -38000000000000 \lor \neg \left(y \leq 155000000000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -29000000000000 \lor \neg \left(y \leq 1100000000000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, z + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -29000000000000.0) (not (<= y 1100000000000.0)))
   (+ 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 <= -29000000000000.0) || !(y <= 1100000000000.0)) {
		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 <= -29000000000000.0) || !(y <= 1100000000000.0))
		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, -29000000000000.0], N[Not[LessEqual[y, 1100000000000.0]], $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 < -2.9e13 or 1.1e12 < 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 47.8

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

   5. Applied rewrites 47.8%

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

   if -2.9e13 < y < 1.1e12

   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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -29000000000000 \lor \neg \left(y \leq 1100000000000\right):\\ \;\;\;\;z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, z + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.9% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 70.9

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

5. Applied rewrites 70.9%

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

6. Final simplification 70.9%

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

Alternative 10: 29.3% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-sin.f64 55.5

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

5. Applied rewrites 55.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 30.8%

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

9. Final simplification 30.8%

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

Reproduce

herbie shell --seed 2024299 
(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))))