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

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. +-commutative 99.9%

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

   3. *-commutative 99.9%

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

   4. distribute-rgt-neg-in 99.9%

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

   5. fma-def 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 3: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \lor \neg \left(z \leq 6.2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(x + 1\right) - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.85) (not (<= z 6.2e-15)))
   (- (+ x 1.0) (* (sin y) z))
   (+ x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.85) || !(z <= 6.2e-15)) {
		tmp = (x + 1.0) - (sin(y) * z);
	} else {
		tmp = x + cos(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 ((z <= (-1.85d0)) .or. (.not. (z <= 6.2d-15))) then
        tmp = (x + 1.0d0) - (sin(y) * z)
    else
        tmp = x + cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.85) || !(z <= 6.2e-15)) {
		tmp = (x + 1.0) - (Math.sin(y) * z);
	} else {
		tmp = x + Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.85) or not (z <= 6.2e-15):
		tmp = (x + 1.0) - (math.sin(y) * z)
	else:
		tmp = x + math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.85) || !(z <= 6.2e-15))
		tmp = Float64(Float64(x + 1.0) - Float64(sin(y) * z));
	else
		tmp = Float64(x + cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.85) || ~((z <= 6.2e-15)))
		tmp = (x + 1.0) - (sin(y) * z);
	else
		tmp = x + cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.85], N[Not[LessEqual[z, 6.2e-15]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8500000000000001 or 6.1999999999999998e-15 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 99.6%

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

   if -1.8500000000000001 < z < 6.1999999999999998e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \lor \neg \left(z \leq 6.2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(x + 1\right) - \sin y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+145} \lor \neg \left(z \leq 9.2 \cdot 10^{+144}\right):\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.22e+145) (not (<= z 9.2e+144)))
   (* z (- (sin y)))
   (+ x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.22e+145) || !(z <= 9.2e+144)) {
		tmp = z * -sin(y);
	} else {
		tmp = x + cos(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 ((z <= (-1.22d+145)) .or. (.not. (z <= 9.2d+144))) then
        tmp = z * -sin(y)
    else
        tmp = x + cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.22e+145) || !(z <= 9.2e+144)) {
		tmp = z * -Math.sin(y);
	} else {
		tmp = x + Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.22e+145) or not (z <= 9.2e+144):
		tmp = z * -math.sin(y)
	else:
		tmp = x + math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.22e+145) || !(z <= 9.2e+144))
		tmp = Float64(z * Float64(-sin(y)));
	else
		tmp = Float64(x + cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.22e+145) || ~((z <= 9.2e+144)))
		tmp = z * -sin(y);
	else
		tmp = x + cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.22e+145], N[Not[LessEqual[z, 9.2e+144]], $MachinePrecision]], N[(z * (-N[Sin[y], $MachinePrecision])), $MachinePrecision], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.21999999999999994e145 or 9.2000000000000006e144 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. neg-mul-1 77.1%

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

      3. *-commutative 77.1%

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

   5. Simplified 77.1%

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

   if -1.21999999999999994e145 < z < 9.2000000000000006e144

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 84.9%

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

      1. +-commutative 84.9%

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

   5. Simplified 84.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+145} \lor \neg \left(z \leq 9.2 \cdot 10^{+144}\right):\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -280 \lor \neg \left(y \leq 2.9 \cdot 10^{-12}\right):\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -280.0) (not (<= y 2.9e-12)))
   (+ x (cos y))
   (+ 1.0 (- x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -280.0) || !(y <= 2.9e-12)) {
		tmp = x + cos(y);
	} else {
		tmp = 1.0 + (x - (y * 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 <= (-280.0d0)) .or. (.not. (y <= 2.9d-12))) then
        tmp = x + cos(y)
    else
        tmp = 1.0d0 + (x - (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -280.0) || !(y <= 2.9e-12)) {
		tmp = x + Math.cos(y);
	} else {
		tmp = 1.0 + (x - (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -280.0) or not (y <= 2.9e-12):
		tmp = x + math.cos(y)
	else:
		tmp = 1.0 + (x - (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -280.0) || !(y <= 2.9e-12))
		tmp = Float64(x + cos(y));
	else
		tmp = Float64(1.0 + Float64(x - Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -280.0) || ~((y <= 2.9e-12)))
		tmp = x + cos(y);
	else
		tmp = 1.0 + (x - (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -280.0], N[Not[LessEqual[y, 2.9e-12]], $MachinePrecision]], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -280 or 2.9000000000000002e-12 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 60.8%

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

      1. +-commutative 60.8%

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

   5. Simplified 60.8%

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

   if -280 < y < 2.9000000000000002e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -280 \lor \neg \left(y \leq 2.9 \cdot 10^{-12}\right):\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x - y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.1% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+17} \lor \neg \left(y \leq 235000\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 + y \cdot \left(y \cdot -0.5 - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.1e+17) (not (<= y 235000.0)))
   (+ x 1.0)
   (+ x (+ 1.0 (* y (- (* y -0.5) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.1e+17) || !(y <= 235000.0)) {
		tmp = x + 1.0;
	} else {
		tmp = x + (1.0 + (y * ((y * -0.5) - 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 <= (-4.1d+17)) .or. (.not. (y <= 235000.0d0))) then
        tmp = x + 1.0d0
    else
        tmp = x + (1.0d0 + (y * ((y * (-0.5d0)) - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.1e+17) || !(y <= 235000.0)) {
		tmp = x + 1.0;
	} else {
		tmp = x + (1.0 + (y * ((y * -0.5) - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.1e+17) or not (y <= 235000.0):
		tmp = x + 1.0
	else:
		tmp = x + (1.0 + (y * ((y * -0.5) - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.1e+17) || !(y <= 235000.0))
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(x + Float64(1.0 + Float64(y * Float64(Float64(y * -0.5) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.1e+17) || ~((y <= 235000.0)))
		tmp = x + 1.0;
	else
		tmp = x + (1.0 + (y * ((y * -0.5) - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.1e+17], N[Not[LessEqual[y, 235000.0]], $MachinePrecision]], N[(x + 1.0), $MachinePrecision], N[(x + N[(1.0 + N[(y * N[(N[(y * -0.5), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1e17 or 235000 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 37.6%

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

      1. +-commutative 37.6%

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

   5. Simplified 37.6%

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

   if -4.1e17 < y < 235000

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. fma-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 96.6%

      \[\leadsto \color{blue}{1 + \left(x + \left(-1 \cdot \left(y \cdot z\right) + -0.5 \cdot {y}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 96.6%

         \[\leadsto \color{blue}{\left(1 + x\right) + \left(-1 \cdot \left(y \cdot z\right) + -0.5 \cdot {y}^{2}\right)} \]

      2. +-commutative 96.6%

         \[\leadsto \color{blue}{\left(x + 1\right)} + \left(-1 \cdot \left(y \cdot z\right) + -0.5 \cdot {y}^{2}\right) \]

      3. associate-+l+ 96.6%

         \[\leadsto \color{blue}{x + \left(1 + \left(-1 \cdot \left(y \cdot z\right) + -0.5 \cdot {y}^{2}\right)\right)} \]

      4. +-commutative 96.6%

         \[\leadsto x + \left(1 + \color{blue}{\left(-0.5 \cdot {y}^{2} + -1 \cdot \left(y \cdot z\right)\right)}\right) \]

      5. mul-1-neg 96.6%

         \[\leadsto x + \left(1 + \left(-0.5 \cdot {y}^{2} + \color{blue}{\left(-y \cdot z\right)}\right)\right) \]

      6. unsub-neg 96.6%

         \[\leadsto x + \left(1 + \color{blue}{\left(-0.5 \cdot {y}^{2} - y \cdot z\right)}\right) \]

      7. *-commutative 96.6%

         \[\leadsto x + \left(1 + \left(\color{blue}{{y}^{2} \cdot -0.5} - y \cdot z\right)\right) \]

      8. unpow2 96.6%

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

      9. associate-*l* 96.6%

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

      10. distribute-lft-out-- 96.6%

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

   7. Simplified 96.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+17} \lor \neg \left(y \leq 235000\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 + y \cdot \left(y \cdot -0.5 - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+35} \lor \neg \left(y \leq 5.5 \cdot 10^{+16}\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.4e+35) (not (<= y 5.5e+16)))
   (+ x 1.0)
   (+ 1.0 (- x (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.4e+35) || !(y <= 5.5e+16)) {
		tmp = x + 1.0;
	} else {
		tmp = 1.0 + (x - (y * 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 <= (-4.4d+35)) .or. (.not. (y <= 5.5d+16))) then
        tmp = x + 1.0d0
    else
        tmp = 1.0d0 + (x - (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.4e+35) || !(y <= 5.5e+16)) {
		tmp = x + 1.0;
	} else {
		tmp = 1.0 + (x - (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.4e+35) or not (y <= 5.5e+16):
		tmp = x + 1.0
	else:
		tmp = 1.0 + (x - (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.4e+35) || !(y <= 5.5e+16))
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(1.0 + Float64(x - Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.4e+35) || ~((y <= 5.5e+16)))
		tmp = x + 1.0;
	else
		tmp = 1.0 + (x - (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.4e+35], N[Not[LessEqual[y, 5.5e+16]], $MachinePrecision]], N[(x + 1.0), $MachinePrecision], N[(1.0 + N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.3999999999999997e35 or 5.5e16 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 38.3%

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

      1. +-commutative 38.3%

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

   5. Simplified 38.3%

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

   if -4.3999999999999997e35 < y < 5.5e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 91.9%

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

      1. mul-1-neg 91.9%

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

      2. unsub-neg 91.9%

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

   5. Simplified 91.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+35} \lor \neg \left(y \leq 5.5 \cdot 10^{+16}\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x - y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.7% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -850000000:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+27}:\\ \;\;\;\;1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -850000000.0) (+ x 1.0) (if (<= x 2.5e+27) (- 1.0 (* y z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -850000000.0) {
		tmp = x + 1.0;
	} else if (x <= 2.5e+27) {
		tmp = 1.0 - (y * z);
	} else {
		tmp = x;
	}
	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 <= (-850000000.0d0)) then
        tmp = x + 1.0d0
    else if (x <= 2.5d+27) then
        tmp = 1.0d0 - (y * z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -850000000.0) {
		tmp = x + 1.0;
	} else if (x <= 2.5e+27) {
		tmp = 1.0 - (y * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -850000000.0:
		tmp = x + 1.0
	elif x <= 2.5e+27:
		tmp = 1.0 - (y * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -850000000.0)
		tmp = Float64(x + 1.0);
	elseif (x <= 2.5e+27)
		tmp = Float64(1.0 - Float64(y * z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -850000000.0)
		tmp = x + 1.0;
	elseif (x <= 2.5e+27)
		tmp = 1.0 - (y * z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -850000000.0], N[(x + 1.0), $MachinePrecision], If[LessEqual[x, 2.5e+27], N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5e8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 89.0%

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

      1. +-commutative 89.0%

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

   5. Simplified 89.0%

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

   if -8.5e8 < x < 2.4999999999999999e27

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 75.9%

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

   4. Taylor expanded in y around 0 44.5%

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

   5. Taylor expanded in x around 0 42.7%

      \[\leadsto \color{blue}{1 - y \cdot z} \]

   if 2.4999999999999999e27 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 82.8%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -850000000:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+27}:\\ \;\;\;\;1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.5% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.02:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.65e-42) x (if (<= x 1.02) 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.65e-42) {
		tmp = x;
	} else if (x <= 1.02) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	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 <= (-2.65d-42)) then
        tmp = x
    else if (x <= 1.02d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.65e-42) {
		tmp = x;
	} else if (x <= 1.02) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.65e-42:
		tmp = x
	elif x <= 1.02:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.65e-42)
		tmp = x;
	elseif (x <= 1.02)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.65e-42)
		tmp = x;
	elseif (x <= 1.02)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.65e-42], x, If[LessEqual[x, 1.02], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.65e-42 or 1.02 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 76.2%

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

   if -2.65e-42 < x < 1.02

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

      5. fma-def 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

      4. neg-mul-1 99.3%

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

      5. fma-def 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in y around 0 34.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.02:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.9% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+243}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.3e+243) (* y (- z)) (+ x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.3e+243) {
		tmp = y * -z;
	} else {
		tmp = x + 1.0;
	}
	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) :: tmp
    if (z <= (-3.3d+243)) then
        tmp = y * -z
    else
        tmp = x + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.3e+243) {
		tmp = y * -z;
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.3e+243:
		tmp = y * -z
	else:
		tmp = x + 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.3e+243)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.3e+243)
		tmp = y * -z;
	else
		tmp = x + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.3e+243], N[(y * (-z)), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.29999999999999994e243

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 99.8%

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

   4. Taylor expanded in y around 0 52.4%

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

   5. Taylor expanded in y around inf 52.4%

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

      1. associate-*r* 52.4%

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

      2. neg-mul-1 52.4%

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

   7. Simplified 52.4%

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

   if -3.29999999999999994e243 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 60.6%

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

      1. +-commutative 60.6%

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

   5. Simplified 60.6%

      \[\leadsto \color{blue}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+243}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.2% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0 58.6%

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

   1. +-commutative 58.6%

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

5. Simplified 58.6%

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

6. Final simplification 58.6%

   \[\leadsto x + 1 \]
7. Add Preprocessing

Alternative 12: 21.1% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. +-commutative 99.9%

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

   3. *-commutative 99.9%

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

   4. distribute-rgt-neg-in 99.9%

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

   5. fma-def 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0 56.2%

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

   1. +-commutative 56.2%

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

   2. associate-*r* 56.2%

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

   3. *-commutative 56.2%

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

   4. neg-mul-1 56.2%

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

   5. fma-def 56.2%

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

7. Simplified 56.2%

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

8. Taylor expanded in y around 0 16.6%

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

9. Final simplification 16.6%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, B"
  :precision binary64
  (- (+ x (cos y)) (* z (sin y))))