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

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + cos(y)) - (z * sin(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 + cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + cos(y)) - (z * sin(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 + cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + cos(y)) - (z * sin(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 + cos(y)) - (z * sin(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - z \cdot \sin y\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9995:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* z (sin y)))))
   (if (<= t_0 -1000000.0)
     (- x (fma y z -1.0))
     (if (<= t_0 0.9995) (cos y) (+ x 1.0)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (z * sin(y));
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = x - fma(y, z, -1.0);
	} else if (t_0 <= 0.9995) {
		tmp = cos(y);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = Float64(x - fma(y, z, -1.0));
	elseif (t_0 <= 0.9995)
		tmp = cos(y);
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], N[(x - N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9995], N[Cos[y], $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1e6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 69.4

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

   5. Applied rewrites 69.4%

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

   if -1e6 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.99950000000000006

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \cos y \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \cos y \]

   if 0.99950000000000006 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x + 1} \]

      2. lower-+.f64 75.8

         \[\leadsto \color{blue}{x + 1} \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{x + 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + 1\right) - z \cdot \sin y\\ \mathbf{if}\;z \leq -3 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.95:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x 1.0) (* z (sin y)))))
   (if (<= z -3e+15) t_0 (if (<= z 0.95) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + 1.0) - (z * sin(y));
	double tmp;
	if (z <= -3e+15) {
		tmp = t_0;
	} else if (z <= 0.95) {
		tmp = x + cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) - (z * sin(y))
    if (z <= (-3d+15)) then
        tmp = t_0
    else if (z <= 0.95d0) then
        tmp = x + cos(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + 1.0) - (z * Math.sin(y));
	double tmp;
	if (z <= -3e+15) {
		tmp = t_0;
	} else if (z <= 0.95) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + 1.0) - (z * math.sin(y))
	tmp = 0
	if z <= -3e+15:
		tmp = t_0
	elif z <= 0.95:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + 1.0) - Float64(z * sin(y)))
	tmp = 0.0
	if (z <= -3e+15)
		tmp = t_0;
	elseif (z <= 0.95)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + 1.0) - (z * sin(y));
	tmp = 0.0;
	if (z <= -3e+15)
		tmp = t_0;
	elseif (z <= 0.95)
		tmp = x + cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3e+15], t$95$0, If[LessEqual[z, 0.95], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3e15 or 0.94999999999999996 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.9%

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

   if -3e15 < z < 0.94999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+15}:\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 0.95:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+118}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+111}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sin y) (- z))))
   (if (<= z -3.5e+118) t_0 (if (<= z 1.6e+111) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) * -z;
	double tmp;
	if (z <= -3.5e+118) {
		tmp = t_0;
	} else if (z <= 1.6e+111) {
		tmp = x + cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) * -z
    if (z <= (-3.5d+118)) then
        tmp = t_0
    else if (z <= 1.6d+111) then
        tmp = x + cos(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) * -z;
	double tmp;
	if (z <= -3.5e+118) {
		tmp = t_0;
	} else if (z <= 1.6e+111) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sin(y) * -z
	tmp = 0
	if z <= -3.5e+118:
		tmp = t_0
	elif z <= 1.6e+111:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) * Float64(-z))
	tmp = 0.0
	if (z <= -3.5e+118)
		tmp = t_0;
	elseif (z <= 1.6e+111)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sin(y) * -z;
	tmp = 0.0;
	if (z <= -3.5e+118)
		tmp = t_0;
	elseif (z <= 1.6e+111)
		tmp = x + cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision]}, If[LessEqual[z, -3.5e+118], t$95$0, If[LessEqual[z, 1.6e+111], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.50000000000000016e118 or 1.6e111 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -3.50000000000000016e118 < z < 1.6e111

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+118}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+111}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ \mathbf{if}\;y \leq -23000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-15}:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y, z \cdot \left(y \cdot -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))))
   (if (<= y -23000.0)
     t_0
     (if (<= y 3.9e-15)
       (- (+ x 1.0) (* y (fma y (* z (* y -0.16666666666666666)) z)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double tmp;
	if (y <= -23000.0) {
		tmp = t_0;
	} else if (y <= 3.9e-15) {
		tmp = (x + 1.0) - (y * fma(y, (z * (y * -0.16666666666666666)), z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	tmp = 0.0
	if (y <= -23000.0)
		tmp = t_0;
	elseif (y <= 3.9e-15)
		tmp = Float64(Float64(x + 1.0) - Float64(y * fma(y, Float64(z * Float64(y * -0.16666666666666666)), z)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -23000.0], t$95$0, If[LessEqual[y, 3.9e-15], N[(N[(x + 1.0), $MachinePrecision] - N[(y * N[(y * N[(z * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -23000 or 3.90000000000000026e-15 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if -23000 < y < 3.90000000000000026e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -23000:\\ \;\;\;\;x + \cos y\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-15}:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y, z \cdot \left(y \cdot -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{-1}{\mathsf{fma}\left(y, -z, -1\right)}\\ \mathbf{elif}\;y \leq 3100000:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.5 - z, x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e+24)
   (+ x (/ -1.0 (fma y (- z) -1.0)))
   (if (<= y 3100000.0) (fma y (- (* y -0.5) z) (+ x 1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+24) {
		tmp = x + (-1.0 / fma(y, -z, -1.0));
	} else if (y <= 3100000.0) {
		tmp = fma(y, ((y * -0.5) - z), (x + 1.0));
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e+24)
		tmp = Float64(x + Float64(-1.0 / fma(y, Float64(-z), -1.0)));
	elseif (y <= 3100000.0)
		tmp = fma(y, Float64(Float64(y * -0.5) - z), Float64(x + 1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e+24], N[(x + N[(-1.0 / N[(y * (-z) + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3100000.0], N[(y * N[(N[(y * -0.5), $MachinePrecision] - z), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.20000000000000002e24

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 31.7

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

   5. Applied rewrites 31.7%

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 40.0%

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

   if -2.20000000000000002e24 < y < 3.1e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if 3.1e6 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x + 1} \]

      2. lower-+.f64 38.4

         \[\leadsto \color{blue}{x + 1} \]

   5. Applied rewrites 38.4%

      \[\leadsto \color{blue}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{-1}{\mathsf{fma}\left(y, -z, -1\right)}\\ \mathbf{elif}\;y \leq 3100000:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.5 - z, x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+24}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 3100000:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot -0.5 - z, x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e+24)
   (+ x 1.0)
   (if (<= y 3100000.0) (fma y (- (* y -0.5) z) (+ x 1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+24) {
		tmp = x + 1.0;
	} else if (y <= 3100000.0) {
		tmp = fma(y, ((y * -0.5) - z), (x + 1.0));
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e+24)
		tmp = Float64(x + 1.0);
	elseif (y <= 3100000.0)
		tmp = fma(y, Float64(Float64(y * -0.5) - z), Float64(x + 1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e+24], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 3100000.0], N[(y * N[(N[(y * -0.5), $MachinePrecision] - z), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000002e24 or 3.1e6 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x + 1} \]

      2. lower-+.f64 39.2

         \[\leadsto \color{blue}{x + 1} \]

   5. Applied rewrites 39.2%

      \[\leadsto \color{blue}{x + 1} \]

   if -2.20000000000000002e24 < y < 3.1e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 96.3

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

   5. Applied rewrites 96.3%

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

Alternative 8: 70.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.24 \cdot 10^{+42}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 1750000:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.24e+42)
   (+ x 1.0)
   (if (<= y 1750000.0) (- x (fma y z -1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.24e+42) {
		tmp = x + 1.0;
	} else if (y <= 1750000.0) {
		tmp = x - fma(y, z, -1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.24e+42)
		tmp = Float64(x + 1.0);
	elseif (y <= 1750000.0)
		tmp = Float64(x - fma(y, z, -1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.24e+42], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 1750000.0], N[(x - N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.24e42 or 1.75e6 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x + 1} \]

      2. lower-+.f64 38.3

         \[\leadsto \color{blue}{x + 1} \]

   5. Applied rewrites 38.3%

      \[\leadsto \color{blue}{x + 1} \]

   if -1.24e42 < y < 1.75e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 95.4

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

   5. Applied rewrites 95.4%

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

Alternative 9: 64.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-19}:\\ \;\;\;\;x - y \cdot z\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.3e-19)
   (- x (* y z))
   (if (<= x 4.2e-9) (fma y (- z) 1.0) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e-19) {
		tmp = x - (y * z);
	} else if (x <= 4.2e-9) {
		tmp = fma(y, -z, 1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.3e-19)
		tmp = Float64(x - Float64(y * z));
	elseif (x <= 4.2e-9)
		tmp = fma(y, Float64(-z), 1.0);
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e-19], N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-9], N[(y * (-z) + 1.0), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000006e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 83.7%

      \[\leadsto x - y \cdot \color{blue}{z} \]

   if -1.30000000000000006e-19 < x < 4.20000000000000039e-9

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 51.7

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{y \cdot z} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.7%

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

   if 4.20000000000000039e-9 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x + 1} \]

      2. lower-+.f64 83.3

         \[\leadsto \color{blue}{x + 1} \]

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{x + 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.2% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0003:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0003) (+ x 1.0) (if (<= x 4.2e-9) (fma y (- z) 1.0) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0003) {
		tmp = x + 1.0;
	} else if (x <= 4.2e-9) {
		tmp = fma(y, -z, 1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.0003)
		tmp = Float64(x + 1.0);
	elseif (x <= 4.2e-9)
		tmp = fma(y, Float64(-z), 1.0);
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.0003], N[(x + 1.0), $MachinePrecision], If[LessEqual[x, 4.2e-9], N[(y * (-z) + 1.0), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999974e-4 or 4.20000000000000039e-9 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x + 1} \]

      2. lower-+.f64 85.6

         \[\leadsto \color{blue}{x + 1} \]

   5. Applied rewrites 85.6%

      \[\leadsto \color{blue}{x + 1} \]

   if -2.99999999999999974e-4 < x < 4.20000000000000039e-9

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{y \cdot z} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.6%

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

Alternative 11: 61.9% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ x + 1 \end{array} \]

FPCore

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

C

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

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 + 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x + 1.0;
}

Python

def code(x, y, z):
	return x + 1.0

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = x + 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x + 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{x + 1} \]

   2. lower-+.f64 63.3

      \[\leadsto \color{blue}{x + 1} \]

5. Applied rewrites 63.3%

   \[\leadsto \color{blue}{x + 1} \]
6. Add Preprocessing

Alternative 12: 21.4% accurate, 212.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{x + 1} \]

   2. lower-+.f64 63.3

      \[\leadsto \color{blue}{x + 1} \]

5. Applied rewrites 63.3%

   \[\leadsto \color{blue}{x + 1} \]

6. Taylor expanded in x around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 21.0%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, B"
  :precision binary64
  (- (+ x (cos y)) (* z (sin y))))