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

Percentage Accurate: 99.9% → 99.9%

Time: 7.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 81.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y + \left(x + \sin y\right)\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;x + z\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z (cos y)) (+ x (sin y)))))
   (if (<= t_0 -40.0)
     (+ x z)
     (if (<= t_0 -0.04)
       (sin y)
       (if (<= t_0 2e-15) (+ (+ x y) z) (if (<= t_0 2.0) (sin y) (+ x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * cos(y)) + (x + sin(y));
	double tmp;
	if (t_0 <= -40.0) {
		tmp = x + z;
	} else if (t_0 <= -0.04) {
		tmp = sin(y);
	} else if (t_0 <= 2e-15) {
		tmp = (x + y) + z;
	} else if (t_0 <= 2.0) {
		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 = (z * cos(y)) + (x + sin(y))
    if (t_0 <= (-40.0d0)) then
        tmp = x + z
    else if (t_0 <= (-0.04d0)) then
        tmp = sin(y)
    else if (t_0 <= 2d-15) then
        tmp = (x + y) + z
    else if (t_0 <= 2.0d0) 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 = (z * Math.cos(y)) + (x + Math.sin(y));
	double tmp;
	if (t_0 <= -40.0) {
		tmp = x + z;
	} else if (t_0 <= -0.04) {
		tmp = Math.sin(y);
	} else if (t_0 <= 2e-15) {
		tmp = (x + y) + z;
	} else if (t_0 <= 2.0) {
		tmp = Math.sin(y);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * math.cos(y)) + (x + math.sin(y))
	tmp = 0
	if t_0 <= -40.0:
		tmp = x + z
	elif t_0 <= -0.04:
		tmp = math.sin(y)
	elif t_0 <= 2e-15:
		tmp = (x + y) + z
	elif t_0 <= 2.0:
		tmp = math.sin(y)
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * cos(y)) + Float64(x + sin(y)))
	tmp = 0.0
	if (t_0 <= -40.0)
		tmp = Float64(x + z);
	elseif (t_0 <= -0.04)
		tmp = sin(y);
	elseif (t_0 <= 2e-15)
		tmp = Float64(Float64(x + y) + z);
	elseif (t_0 <= 2.0)
		tmp = sin(y);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * cos(y)) + (x + sin(y));
	tmp = 0.0;
	if (t_0 <= -40.0)
		tmp = x + z;
	elseif (t_0 <= -0.04)
		tmp = sin(y);
	elseif (t_0 <= 2e-15)
		tmp = (x + y) + z;
	elseif (t_0 <= 2.0)
		tmp = sin(y);
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -40.0], N[(x + z), $MachinePrecision], If[LessEqual[t$95$0, -0.04], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 2e-15], N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 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))) < -40 or 2 < (+.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 77.1

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

   5. Applied rewrites 77.1%

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

   if -40 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.0400000000000000008 or 2.0000000000000002e-15 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 2

   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.1%

      \[\leadsto \sin y \]

   if -0.0400000000000000008 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 2.0000000000000002e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \cos y + \left(x + \sin y\right) \leq -40:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \cdot \cos y + \left(x + \sin y\right) \leq -0.04:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;z \cdot \cos y + \left(x + \sin y\right) \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{elif}\;z \cdot \cos y + \left(x + \sin y\right) \leq 2:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.28:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.28)
   (fma (cos y) z (sin y))
   (if (<= y 1.8e+140) (fma (cos y) z (+ x y)) (+ x (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.28) {
		tmp = fma(cos(y), z, sin(y));
	} else if (y <= 1.8e+140) {
		tmp = fma(cos(y), z, (x + y));
	} else {
		tmp = x + sin(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.28)
		tmp = fma(cos(y), z, sin(y));
	elseif (y <= 1.8e+140)
		tmp = fma(cos(y), z, Float64(x + y));
	else
		tmp = Float64(x + sin(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.28], N[(N[Cos[y], $MachinePrecision] * z + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+140], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.28000000000000003

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if -0.28000000000000003 < y < 1.8e140

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 95.6

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

   7. Applied rewrites 95.6%

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

   if 1.8e140 < y

   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 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.28:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -4.7 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-45}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-16}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+206}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -4.7e+129)
     t_0
     (if (<= z -2e-45)
       (+ x z)
       (if (<= z 2.9e-16) (+ x (sin y)) (if (<= z 2.8e+206) (+ x z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -4.7e+129) {
		tmp = t_0;
	} else if (z <= -2e-45) {
		tmp = x + z;
	} else if (z <= 2.9e-16) {
		tmp = x + sin(y);
	} else if (z <= 2.8e+206) {
		tmp = x + z;
	} 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 <= (-4.7d+129)) then
        tmp = t_0
    else if (z <= (-2d-45)) then
        tmp = x + z
    else if (z <= 2.9d-16) then
        tmp = x + sin(y)
    else if (z <= 2.8d+206) then
        tmp = x + z
    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 <= -4.7e+129) {
		tmp = t_0;
	} else if (z <= -2e-45) {
		tmp = x + z;
	} else if (z <= 2.9e-16) {
		tmp = x + Math.sin(y);
	} else if (z <= 2.8e+206) {
		tmp = x + z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -4.7e+129:
		tmp = t_0
	elif z <= -2e-45:
		tmp = x + z
	elif z <= 2.9e-16:
		tmp = x + math.sin(y)
	elif z <= 2.8e+206:
		tmp = x + z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -4.7e+129)
		tmp = t_0;
	elseif (z <= -2e-45)
		tmp = Float64(x + z);
	elseif (z <= 2.9e-16)
		tmp = Float64(x + sin(y));
	elseif (z <= 2.8e+206)
		tmp = Float64(x + z);
	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 <= -4.7e+129)
		tmp = t_0;
	elseif (z <= -2e-45)
		tmp = x + z;
	elseif (z <= 2.9e-16)
		tmp = x + sin(y);
	elseif (z <= 2.8e+206)
		tmp = x + z;
	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, -4.7e+129], t$95$0, If[LessEqual[z, -2e-45], N[(x + z), $MachinePrecision], If[LessEqual[z, 2.9e-16], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.8e+206], N[(x + z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.70000000000000008e129 or 2.7999999999999998e206 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -4.70000000000000008e129 < z < -1.99999999999999997e-45 or 2.8999999999999998e-16 < z < 2.7999999999999998e206

   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 78.4

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

   5. Applied rewrites 78.4%

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

   if -1.99999999999999997e-45 < z < 2.8999999999999998e-16

   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 94.6

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

   5. Applied rewrites 94.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+129}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-45}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-16}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+206}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+129}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-45}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+67}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.7e+129)
   (* z (cos y))
   (if (<= z -2e-45)
     (+ x z)
     (if (<= z 3.3e+67) (+ x (sin y)) (fma (cos y) z (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.7e+129) {
		tmp = z * cos(y);
	} else if (z <= -2e-45) {
		tmp = x + z;
	} else if (z <= 3.3e+67) {
		tmp = x + sin(y);
	} else {
		tmp = fma(cos(y), z, (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.7e+129)
		tmp = Float64(z * cos(y));
	elseif (z <= -2e-45)
		tmp = Float64(x + z);
	elseif (z <= 3.3e+67)
		tmp = Float64(x + sin(y));
	else
		tmp = fma(cos(y), z, Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.7e+129], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2e-45], N[(x + z), $MachinePrecision], If[LessEqual[z, 3.3e+67], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.70000000000000008e129

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if -4.70000000000000008e129 < z < -1.99999999999999997e-45

   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 97.1

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

   5. Applied rewrites 97.1%

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

   if -1.99999999999999997e-45 < z < 3.3000000000000003e67

   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 88.9

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

   5. Applied rewrites 88.9%

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

   if 3.3000000000000003e67 < z

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 84.5

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

   7. Applied rewrites 84.5%

      \[\leadsto \mathsf{fma}\left(\cos y, z, \color{blue}{y + x}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+129}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-45}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+67}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sin y\\ \mathbf{if}\;y \leq -0.00016:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 520000:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (sin y))))
   (if (<= y -0.00016) t_0 (if (<= y 520000.0) (+ (+ x y) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + sin(y);
	double tmp;
	if (y <= -0.00016) {
		tmp = t_0;
	} else if (y <= 520000.0) {
		tmp = (x + y) + z;
	} 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 + sin(y)
    if (y <= (-0.00016d0)) then
        tmp = t_0
    else if (y <= 520000.0d0) then
        tmp = (x + y) + z
    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 + Math.sin(y);
	double tmp;
	if (y <= -0.00016) {
		tmp = t_0;
	} else if (y <= 520000.0) {
		tmp = (x + y) + z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + math.sin(y)
	tmp = 0
	if y <= -0.00016:
		tmp = t_0
	elif y <= 520000.0:
		tmp = (x + y) + z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + sin(y))
	tmp = 0.0
	if (y <= -0.00016)
		tmp = t_0;
	elseif (y <= 520000.0)
		tmp = Float64(Float64(x + y) + z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + sin(y);
	tmp = 0.0;
	if (y <= -0.00016)
		tmp = t_0;
	elseif (y <= 520000.0)
		tmp = (x + y) + z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00016], t$95$0, If[LessEqual[y, 520000.0], N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.60000000000000013e-4 or 5.2e5 < 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 62.1

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

   5. Applied rewrites 62.1%

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

   if -1.60000000000000013e-4 < y < 5.2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00016:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;y \leq 520000:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+38}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.05e+38)
   (+ x z)
   (if (<= y 1.4e-6)
     (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y (+ x z))
     (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.05e+38) {
		tmp = x + z;
	} else if (y <= 1.4e-6) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.05e+38)
		tmp = Float64(x + z);
	elseif (y <= 1.4e-6)
		tmp = fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.05e+38], N[(x + z), $MachinePrecision], If[LessEqual[y, 1.4e-6], N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e38 or 1.39999999999999994e-6 < 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 40.7

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

   5. Applied rewrites 40.7%

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

   if -1.05e38 < y < 1.39999999999999994e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+38}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+38}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.05e+38)
   (+ x z)
   (if (<= y 1.4e-6) (fma (fma (* -0.5 y) z 1.0) y (+ x z)) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.05e+38) {
		tmp = x + z;
	} else if (y <= 1.4e-6) {
		tmp = fma(fma((-0.5 * y), z, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.05e+38)
		tmp = Float64(x + z);
	elseif (y <= 1.4e-6)
		tmp = fma(fma(Float64(-0.5 * y), z, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.05e+38], N[(x + z), $MachinePrecision], If[LessEqual[y, 1.4e-6], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e38 or 1.39999999999999994e-6 < 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 40.7

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

   5. Applied rewrites 40.7%

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

   if -1.05e38 < y < 1.39999999999999994e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 95.7

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

   5. Applied rewrites 95.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+38}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.4% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+67}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-27}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.9e+67) (+ x z) (if (<= y 6e-27) (+ (+ x y) z) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.9e+67) {
		tmp = x + z;
	} else if (y <= 6e-27) {
		tmp = (x + y) + z;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.9d+67)) then
        tmp = x + z
    else if (y <= 6d-27) then
        tmp = (x + y) + z
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.9e+67) {
		tmp = x + z;
	} else if (y <= 6e-27) {
		tmp = (x + y) + z;
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.9e+67:
		tmp = x + z
	elif y <= 6e-27:
		tmp = (x + y) + z
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.9e+67)
		tmp = Float64(x + z);
	elseif (y <= 6e-27)
		tmp = Float64(Float64(x + y) + z);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.9e+67)
		tmp = x + z;
	elseif (y <= 6e-27)
		tmp = (x + y) + z;
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.9e+67], N[(x + z), $MachinePrecision], If[LessEqual[y, 6e-27], N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8999999999999999e67 or 6.0000000000000002e-27 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 42.8

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

   5. Applied rewrites 42.8%

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

   if -4.8999999999999999e67 < y < 6.0000000000000002e-27

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 95.0

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

   5. Applied rewrites 95.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+67}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-27}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.1% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-21}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+60}:\\ \;\;\;\;z + y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.15e-21) (+ x y) (if (<= x 1.2e+60) (+ z y) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.15e-21) {
		tmp = x + y;
	} else if (x <= 1.2e+60) {
		tmp = z + y;
	} 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 <= (-2.15d-21)) then
        tmp = x + y
    else if (x <= 1.2d+60) then
        tmp = z + y
    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 <= -2.15e-21) {
		tmp = x + y;
	} else if (x <= 1.2e+60) {
		tmp = z + y;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.15e-21:
		tmp = x + y
	elif x <= 1.2e+60:
		tmp = z + y
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.15e-21)
		tmp = Float64(x + y);
	elseif (x <= 1.2e+60)
		tmp = Float64(z + y);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.15e-21)
		tmp = x + y;
	elseif (x <= 1.2e+60)
		tmp = z + y;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.15e-21], N[(x + y), $MachinePrecision], If[LessEqual[x, 1.2e+60], N[(z + y), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1499999999999999e-21 or 1.2e60 < 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

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.3%

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

   if -2.1499999999999999e-21 < x < 1.2e60

   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 45.2

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 51.9

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

   8. Applied rewrites 51.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 42.7%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-21}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+60}:\\ \;\;\;\;z + y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.8% 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 66.5

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

5. Applied rewrites 66.5%

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

6. Final simplification 66.5%

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

Alternative 12: 38.2% 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

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 57.9

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

5. Applied rewrites 57.9%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 43.4%

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

9. Final simplification 43.4%

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

Reproduce

herbie shell --seed 2024249 
(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))))