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

Percentage Accurate: 99.9% → 99.9%

Time: 9.7s

Alternatives: 14

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

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

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]

Derivation

1. Initial program 99.9%

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

Alternative 2: 81.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -500000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;y + \left(x + z\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (<= t_0 -500000.0)
     (+ x z)
     (if (<= t_0 -0.01)
       (sin y)
       (if (<= t_0 2e-13) (+ y (+ x z)) (if (<= t_0 1.0) (sin 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 <= -500000.0) {
		tmp = x + z;
	} else if (t_0 <= -0.01) {
		tmp = sin(y);
	} else if (t_0 <= 2e-13) {
		tmp = y + (x + z);
	} else if (t_0 <= 1.0) {
		tmp = sin(y);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x + sin(y)) + (z * cos(y))
    if (t_0 <= (-500000.0d0)) then
        tmp = x + z
    else if (t_0 <= (-0.01d0)) then
        tmp = sin(y)
    else if (t_0 <= 2d-13) then
        tmp = y + (x + z)
    else if (t_0 <= 1.0d0) then
        tmp = sin(y)
    else
        tmp = 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 <= -500000.0) {
		tmp = x + z;
	} else if (t_0 <= -0.01) {
		tmp = Math.sin(y);
	} else if (t_0 <= 2e-13) {
		tmp = y + (x + z);
	} else if (t_0 <= 1.0) {
		tmp = Math.sin(y);
	} else {
		tmp = 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 <= -500000.0:
		tmp = x + z
	elif t_0 <= -0.01:
		tmp = math.sin(y)
	elif t_0 <= 2e-13:
		tmp = y + (x + z)
	elif t_0 <= 1.0:
		tmp = math.sin(y)
	else:
		tmp = 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 <= -500000.0)
		tmp = Float64(x + z);
	elseif (t_0 <= -0.01)
		tmp = sin(y);
	elseif (t_0 <= 2e-13)
		tmp = Float64(y + Float64(x + z));
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = Float64(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 <= -500000.0)
		tmp = x + z;
	elseif (t_0 <= -0.01)
		tmp = sin(y);
	elseif (t_0 <= 2e-13)
		tmp = y + (x + z);
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = 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[LessEqual[t$95$0, -500000.0], N[(x + z), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 2e-13], N[(y + N[(x + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[y], $MachinePrecision], N[(x + z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -5e5 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -5e5 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.0100000000000000002 or 2.0000000000000001e-13 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 97.4%

      \[\leadsto \sin y \]

   if -0.0100000000000000002 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 2.0000000000000001e-13

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + z \cdot \cos y \leq -500000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq -0.01:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;y + \left(x + z\right)\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \sin y\right) + z \cdot \cos y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;t\_0 \leq 0.3:\\ \;\;\;\;y + \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (+ x (sin y)) (* z (cos y)))))
   (if (<= t_0 -5e+51) (+ x z) (if (<= t_0 0.3) (+ y (+ x z)) (+ 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 <= -5e+51) {
		tmp = x + z;
	} else if (t_0 <= 0.3) {
		tmp = y + (x + 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) :: t_0
    real(8) :: tmp
    t_0 = (x + sin(y)) + (z * cos(y))
    if (t_0 <= (-5d+51)) then
        tmp = x + z
    else if (t_0 <= 0.3d0) then
        tmp = y + (x + z)
    else
        tmp = 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 <= -5e+51) {
		tmp = x + z;
	} else if (t_0 <= 0.3) {
		tmp = y + (x + z);
	} else {
		tmp = 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 <= -5e+51:
		tmp = x + z
	elif t_0 <= 0.3:
		tmp = y + (x + z)
	else:
		tmp = 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 <= -5e+51)
		tmp = Float64(x + z);
	elseif (t_0 <= 0.3)
		tmp = Float64(y + Float64(x + z));
	else
		tmp = Float64(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 <= -5e+51)
		tmp = x + z;
	elseif (t_0 <= 0.3)
		tmp = y + (x + z);
	else
		tmp = 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[LessEqual[t$95$0, -5e+51], N[(x + z), $MachinePrecision], If[LessEqual[t$95$0, 0.3], N[(y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -5e51 or 0.299999999999999989 < (+.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 71.3

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

   5. Applied rewrites 71.3%

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

   if -5e51 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 0.299999999999999989

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 69.7

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

   5. Applied rewrites 69.7%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \sin y\right) + z \cdot \cos y \leq -5 \cdot 10^{+51}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;\left(x + \sin y\right) + z \cdot \cos y \leq 0.3:\\ \;\;\;\;y + \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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

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

4. Applied rewrites 99.9%

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

Alternative 5: 82.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-136}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 8000000:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -1.3e+76)
     t_0
     (if (<= z -4.8e-136) (+ x z) (if (<= z 8000000.0) (+ x (sin y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -1.3e+76) {
		tmp = t_0;
	} else if (z <= -4.8e-136) {
		tmp = x + z;
	} else if (z <= 8000000.0) {
		tmp = x + sin(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 = z * cos(y)
    if (z <= (-1.3d+76)) then
        tmp = t_0
    else if (z <= (-4.8d-136)) then
        tmp = x + z
    else if (z <= 8000000.0d0) then
        tmp = x + sin(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 = z * Math.cos(y);
	double tmp;
	if (z <= -1.3e+76) {
		tmp = t_0;
	} else if (z <= -4.8e-136) {
		tmp = x + z;
	} else if (z <= 8000000.0) {
		tmp = x + Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -1.3e+76:
		tmp = t_0
	elif z <= -4.8e-136:
		tmp = x + z
	elif z <= 8000000.0:
		tmp = x + math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -1.3e+76)
		tmp = t_0;
	elseif (z <= -4.8e-136)
		tmp = Float64(x + z);
	elseif (z <= 8000000.0)
		tmp = Float64(x + sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -1.3e+76)
		tmp = t_0;
	elseif (z <= -4.8e-136)
		tmp = x + z;
	elseif (z <= 8000000.0)
		tmp = x + sin(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.3e+76], t$95$0, If[LessEqual[z, -4.8e-136], N[(x + z), $MachinePrecision], If[LessEqual[z, 8000000.0], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e76 or 8e6 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -1.3e76 < z < -4.7999999999999997e-136

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if -4.7999999999999997e-136 < z < 8e6

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+76}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-136}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 8000000:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+16}:\\ \;\;\;\;\left(x + \sin y\right) + z \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -1.3e+76) t_0 (if (<= z 3e+16) (+ (+ x (sin y)) (* z 1.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -1.3e+76) {
		tmp = t_0;
	} else if (z <= 3e+16) {
		tmp = (x + sin(y)) + (z * 1.0);
	} 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 = z * cos(y)
    if (z <= (-1.3d+76)) then
        tmp = t_0
    else if (z <= 3d+16) then
        tmp = (x + sin(y)) + (z * 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (z <= -1.3e+76) {
		tmp = t_0;
	} else if (z <= 3e+16) {
		tmp = (x + Math.sin(y)) + (z * 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -1.3e+76:
		tmp = t_0
	elif z <= 3e+16:
		tmp = (x + math.sin(y)) + (z * 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -1.3e+76)
		tmp = t_0;
	elseif (z <= 3e+16)
		tmp = Float64(Float64(x + sin(y)) + Float64(z * 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -1.3e+76)
		tmp = t_0;
	elseif (z <= 3e+16)
		tmp = (x + sin(y)) + (z * 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.3e+76], t$95$0, If[LessEqual[z, 3e+16], N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e76 or 3e16 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -1.3e76 < z < 3e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.5%

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

Alternative 7: 80.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sin y\\ \mathbf{if}\;y \leq -0.22:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), 1\right), z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (sin y))))
   (if (<= y -0.22)
     t_0
     (if (<= y 1.95e+15)
       (fma
        (fma y (* y (fma (* y y) 0.041666666666666664 -0.5)) 1.0)
        z
        (+ x y))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + sin(y);
	double tmp;
	if (y <= -0.22) {
		tmp = t_0;
	} else if (y <= 1.95e+15) {
		tmp = fma(fma(y, (y * fma((y * y), 0.041666666666666664, -0.5)), 1.0), z, (x + y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + sin(y))
	tmp = 0.0
	if (y <= -0.22)
		tmp = t_0;
	elseif (y <= 1.95e+15)
		tmp = fma(fma(y, Float64(y * fma(Float64(y * y), 0.041666666666666664, -0.5)), 1.0), z, Float64(x + y));
	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, -0.22], t$95$0, If[LessEqual[y, 1.95e+15], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.220000000000000001 or 1.95e15 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 61.3

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

   5. Applied rewrites 61.3%

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

   if -0.220000000000000001 < y < 1.95e15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.2

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

   10. Applied rewrites 99.2%

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

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z, x + y\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 99.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.22:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), 1\right), z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+16}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right), x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.16e+16)
   (+ x z)
   (if (<= y 3.1e-40)
     (+
      (fma y (fma y (* y -0.16666666666666666) 1.0) x)
      (*
       z
       (fma
        (* y y)
        (fma
         (* y y)
         (fma y (* y -0.001388888888888889) 0.041666666666666664)
         -0.5)
        1.0)))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.16e+16) {
		tmp = x + z;
	} else if (y <= 3.1e-40) {
		tmp = fma(y, fma(y, (y * -0.16666666666666666), 1.0), x) + (z * fma((y * y), fma((y * y), fma(y, (y * -0.001388888888888889), 0.041666666666666664), -0.5), 1.0));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.16e+16)
		tmp = Float64(x + z);
	elseif (y <= 3.1e-40)
		tmp = Float64(fma(y, fma(y, Float64(y * -0.16666666666666666), 1.0), x) + Float64(z * fma(Float64(y * y), fma(Float64(y * y), fma(y, Float64(y * -0.001388888888888889), 0.041666666666666664), -0.5), 1.0)));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.16e+16], N[(x + z), $MachinePrecision], If[LessEqual[y, 3.1e-40], N[(N[(y * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision] + N[(z * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.16e16 or 3.10000000000000011e-40 < 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 45.1

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

   5. Applied rewrites 45.1%

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

   if -1.16e16 < y < 3.10000000000000011e-40

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + x\right)} + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1 + \frac{-1}{6} \cdot {y}^{2}, x\right)} + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{6} \cdot {y}^{2} + 1}, x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{6} \cdot \color{blue}{\left(y \cdot y\right)} + 1, x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{-1}{6} \cdot y\right) \cdot y} + 1, x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{-1}{6} \cdot y\right)} + 1, x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{-1}{6} \cdot y, 1\right)}, x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{6}}, 1\right), x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \]

       9. lower-*.f64 98.7

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot -0.16666666666666666}, 1\right), x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \]

   11. Applied rewrites 98.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right), x\right)} + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+16}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right), x\right) + z \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -400:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), 1\right), z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -400.0)
   (+ x z)
   (if (<= y 3.1e-40)
     (fma (fma y (* y (fma (* y y) 0.041666666666666664 -0.5)) 1.0) z (+ x y))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -400.0) {
		tmp = x + z;
	} else if (y <= 3.1e-40) {
		tmp = fma(fma(y, (y * fma((y * y), 0.041666666666666664, -0.5)), 1.0), z, (x + y));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -400.0)
		tmp = Float64(x + z);
	elseif (y <= 3.1e-40)
		tmp = fma(fma(y, Float64(y * fma(Float64(y * y), 0.041666666666666664, -0.5)), 1.0), z, Float64(x + y));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -400.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 3.1e-40], N[(N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -400 or 3.10000000000000011e-40 < 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 44.8

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

   5. Applied rewrites 44.8%

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

   if -400 < y < 3.10000000000000011e-40

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.4

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

   8. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.4

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

   10. Applied rewrites 99.4%

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

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24}, \frac{-1}{2}\right), 1\right), z, x + y\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 99.4%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -400:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664, -0.5\right), 1\right), z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+16}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, z \cdot -0.5\right), 1\right), x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.16e+16)
   (+ x z)
   (if (<= y 3.1e-40)
     (fma y (fma y (fma y -0.16666666666666666 (* z -0.5)) 1.0) (+ x z))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.16e+16) {
		tmp = x + z;
	} else if (y <= 3.1e-40) {
		tmp = fma(y, fma(y, fma(y, -0.16666666666666666, (z * -0.5)), 1.0), (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.16e+16)
		tmp = Float64(x + z);
	elseif (y <= 3.1e-40)
		tmp = fma(y, fma(y, fma(y, -0.16666666666666666, Float64(z * -0.5)), 1.0), Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.16e+16], N[(x + z), $MachinePrecision], If[LessEqual[y, 3.1e-40], N[(y * N[(y * N[(y * -0.16666666666666666 + N[(z * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.16e16 or 3.10000000000000011e-40 < 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 45.1

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

   5. Applied rewrites 45.1%

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

   if -1.16e16 < y < 3.10000000000000011e-40

   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. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+16}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.16666666666666666, z \cdot -0.5\right), 1\right), x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -680:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-40}:\\ \;\;\;\;z + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot -0.5, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -680.0)
   (+ x z)
   (if (<= y 3.1e-40) (+ z (fma y (fma y (* z -0.5) 1.0) x)) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -680.0) {
		tmp = x + z;
	} else if (y <= 3.1e-40) {
		tmp = z + fma(y, fma(y, (z * -0.5), 1.0), x);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -680.0)
		tmp = Float64(x + z);
	elseif (y <= 3.1e-40)
		tmp = Float64(z + fma(y, fma(y, Float64(z * -0.5), 1.0), x));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -680.0], N[(x + z), $MachinePrecision], If[LessEqual[y, 3.1e-40], N[(z + N[(y * N[(y * N[(z * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -680 or 3.10000000000000011e-40 < 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 44.8

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

   5. Applied rewrites 44.8%

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

   if -680 < y < 3.10000000000000011e-40

   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

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -680:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-40}:\\ \;\;\;\;z + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot -0.5, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.2% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+150}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{-63}:\\ \;\;\;\;y + z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6e+150) (+ x y) (if (<= x 9.4e-63) (+ y z) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e+150) {
		tmp = x + y;
	} else if (x <= 9.4e-63) {
		tmp = y + z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.6d+150)) then
        tmp = x + y
    else if (x <= 9.4d-63) then
        tmp = y + z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e+150) {
		tmp = x + y;
	} else if (x <= 9.4e-63) {
		tmp = y + z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.6e+150:
		tmp = x + y
	elif x <= 9.4e-63:
		tmp = y + z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.6e+150)
		tmp = Float64(x + y);
	elseif (x <= 9.4e-63)
		tmp = Float64(y + z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.6e+150)
		tmp = x + y;
	elseif (x <= 9.4e-63)
		tmp = y + z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.6e+150], N[(x + y), $MachinePrecision], If[LessEqual[x, 9.4e-63], N[(y + z), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000008e150 or 9.4000000000000001e-63 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 57.9%

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

   if -1.60000000000000008e150 < x < 9.4000000000000001e-63

   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. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 47.4%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+150}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{-63}:\\ \;\;\;\;y + z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.2% 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 65.6

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

5. Applied rewrites 65.6%

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

6. Final simplification 65.6%

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

Alternative 14: 37.4% accurate, 53.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + 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 + y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 61.4

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

5. Applied rewrites 61.4%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 36.4%

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

Reproduce

herbie shell --seed 2024221 
(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))))