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

Percentage Accurate: 99.9% → 99.9%

Time: 12.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 84.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+168} \lor \neg \left(z \leq -4.5 \cdot 10^{+122} \lor \neg \left(z \leq -1.7 \cdot 10^{+84}\right) \land z \leq 3.2 \cdot 10^{+79}\right):\\ \;\;\;\;1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.2e+168)
         (not
          (or (<= z -4.5e+122) (and (not (<= z -1.7e+84)) (<= z 3.2e+79)))))
   (- 1.0 (* z (sin y)))
   (+ x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.2e+168) || !((z <= -4.5e+122) || (!(z <= -1.7e+84) && (z <= 3.2e+79)))) {
		tmp = 1.0 - (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 <= (-1.2d+168)) .or. (.not. (z <= (-4.5d+122)) .or. (.not. (z <= (-1.7d+84))) .and. (z <= 3.2d+79))) then
        tmp = 1.0d0 - (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 <= -1.2e+168) || !((z <= -4.5e+122) || (!(z <= -1.7e+84) && (z <= 3.2e+79)))) {
		tmp = 1.0 - (z * Math.sin(y));
	} else {
		tmp = x + Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.2e+168) or not ((z <= -4.5e+122) or (not (z <= -1.7e+84) and (z <= 3.2e+79))):
		tmp = 1.0 - (z * math.sin(y))
	else:
		tmp = x + math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.2e+168) || !((z <= -4.5e+122) || (!(z <= -1.7e+84) && (z <= 3.2e+79))))
		tmp = Float64(1.0 - Float64(z * 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 <= -1.2e+168) || ~(((z <= -4.5e+122) || (~((z <= -1.7e+84)) && (z <= 3.2e+79)))))
		tmp = 1.0 - (z * sin(y));
	else
		tmp = x + cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.2e+168], N[Not[Or[LessEqual[z, -4.5e+122], And[N[Not[LessEqual[z, -1.7e+84]], $MachinePrecision], LessEqual[z, 3.2e+79]]]], $MachinePrecision]], N[(1.0 - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.20000000000000005e168 or -4.49999999999999997e122 < z < -1.6999999999999999e84 or 3.20000000000000003e79 < z

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. add-cube-cbrt 98.7%

         \[\leadsto \left(-\color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \sqrt[3]{z \cdot \sin y}}\right) + \left(x + \cos y\right) \]

      4. distribute-rgt-neg-in 98.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \left(-\sqrt[3]{z \cdot \sin y}\right)} + \left(x + \cos y\right) \]

      5. fma-define 98.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

      6. pow2 98.7%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right) \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

   5. Taylor expanded in x around inf 78.8%

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

      1. fma-define 78.8%

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

      2. associate-/l* 78.7%

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

   7. Simplified 78.7%

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

   8. Taylor expanded in y around 0 78.7%

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

   9. Taylor expanded in x around 0 78.1%

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

       1. mul-1-neg 78.1%

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

       2. unsub-neg 78.1%

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

   11. Simplified 78.1%

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

   if -1.20000000000000005e168 < z < -4.49999999999999997e122 or -1.6999999999999999e84 < z < 3.20000000000000003e79

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 93.6%

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

      1. +-commutative 93.6%

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

   5. Simplified 93.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+168} \lor \neg \left(z \leq -4.5 \cdot 10^{+122} \lor \neg \left(z \leq -1.7 \cdot 10^{+84}\right) \land z \leq 3.2 \cdot 10^{+79}\right):\\ \;\;\;\;1 - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+168} \lor \neg \left(z \leq -1.7 \cdot 10^{+123} \lor \neg \left(z \leq -2.8 \cdot 10^{+82}\right) \land z \leq 6.8 \cdot 10^{+94}\right):\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.22e+168)
         (not
          (or (<= z -1.7e+123) (and (not (<= z -2.8e+82)) (<= z 6.8e+94)))))
   (* z (- (sin y)))
   (+ x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.22e+168) || !((z <= -1.7e+123) || (!(z <= -2.8e+82) && (z <= 6.8e+94)))) {
		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+168)) .or. (.not. (z <= (-1.7d+123)) .or. (.not. (z <= (-2.8d+82))) .and. (z <= 6.8d+94))) 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+168) || !((z <= -1.7e+123) || (!(z <= -2.8e+82) && (z <= 6.8e+94)))) {
		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+168) or not ((z <= -1.7e+123) or (not (z <= -2.8e+82) and (z <= 6.8e+94))):
		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+168) || !((z <= -1.7e+123) || (!(z <= -2.8e+82) && (z <= 6.8e+94))))
		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+168) || ~(((z <= -1.7e+123) || (~((z <= -2.8e+82)) && (z <= 6.8e+94)))))
		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+168], N[Not[Or[LessEqual[z, -1.7e+123], And[N[Not[LessEqual[z, -2.8e+82]], $MachinePrecision], LessEqual[z, 6.8e+94]]]], $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 < -1.21999999999999991e168 or -1.70000000000000001e123 < z < -2.8e82 or 6.8000000000000004e94 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf 71.7%

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

      1. associate-*r* 71.7%

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

      2. neg-mul-1 71.7%

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

      3. *-commutative 71.7%

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

   5. Simplified 71.7%

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

   if -1.21999999999999991e168 < z < -1.70000000000000001e123 or -2.8e82 < z < 6.8000000000000004e94

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 91.9%

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

      1. +-commutative 91.9%

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

   5. Simplified 91.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+168} \lor \neg \left(z \leq -1.7 \cdot 10^{+123} \lor \neg \left(z \leq -2.8 \cdot 10^{+82}\right) \land z \leq 6.8 \cdot 10^{+94}\right):\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.55 \lor \neg \left(z \leq 0.2\right):\\ \;\;\;\;1 + \left(x - z \cdot \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.55) (not (<= z 0.2)))
   (+ 1.0 (- x (* z (sin y))))
   (+ x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.55) || !(z <= 0.2)) {
		tmp = 1.0 + (x - (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 <= (-0.55d0)) .or. (.not. (z <= 0.2d0))) then
        tmp = 1.0d0 + (x - (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 <= -0.55) || !(z <= 0.2)) {
		tmp = 1.0 + (x - (z * Math.sin(y)));
	} else {
		tmp = x + Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -0.55) or not (z <= 0.2):
		tmp = 1.0 + (x - (z * math.sin(y)))
	else:
		tmp = x + math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.55) || !(z <= 0.2))
		tmp = Float64(1.0 + Float64(x - Float64(z * 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 <= -0.55) || ~((z <= 0.2)))
		tmp = 1.0 + (x - (z * sin(y)));
	else
		tmp = x + cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.55], N[Not[LessEqual[z, 0.2]], $MachinePrecision]], N[(1.0 + N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.55000000000000004 or 0.20000000000000001 < z

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. add-cube-cbrt 98.9%

         \[\leadsto \left(-\color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \sqrt[3]{z \cdot \sin y}}\right) + \left(x + \cos y\right) \]

      4. distribute-rgt-neg-in 98.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \left(-\sqrt[3]{z \cdot \sin y}\right)} + \left(x + \cos y\right) \]

      5. fma-define 98.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

      6. pow2 98.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right) \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

   5. Taylor expanded in x around inf 84.2%

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

      1. fma-define 84.2%

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

      2. associate-/l* 84.1%

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

   7. Simplified 84.1%

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

   8. Taylor expanded in y around 0 84.0%

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

   9. Taylor expanded in x around 0 99.7%

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

       1. associate-*r* 99.7%

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

       2. mul-1-neg 99.7%

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

   11. Simplified 99.7%

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

   if -0.55000000000000004 < z < 0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 99.3%

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

      1. +-commutative 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.55 \lor \neg \left(z \leq 0.2\right):\\ \;\;\;\;1 + \left(x - z \cdot \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -12500000000000 \lor \neg \left(y \leq 3.6 \cdot 10^{-15}\right):\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -12500000000000.0) (not (<= y 3.6e-15)))
   (+ x (cos y))
   (+ 1.0 (+ x (* y (- (* y (- (* 0.16666666666666666 (* y z)) 0.5)) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -12500000000000.0) || !(y <= 3.6e-15)) {
		tmp = x + cos(y);
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 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 <= (-12500000000000.0d0)) .or. (.not. (y <= 3.6d-15))) then
        tmp = x + cos(y)
    else
        tmp = 1.0d0 + (x + (y * ((y * ((0.16666666666666666d0 * (y * z)) - 0.5d0)) - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -12500000000000.0) || !(y <= 3.6e-15)) {
		tmp = x + Math.cos(y);
	} else {
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -12500000000000.0) or not (y <= 3.6e-15):
		tmp = x + math.cos(y)
	else:
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -12500000000000.0) || !(y <= 3.6e-15))
		tmp = Float64(x + cos(y));
	else
		tmp = Float64(1.0 + Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(0.16666666666666666 * Float64(y * z)) - 0.5)) - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -12500000000000.0) || ~((y <= 3.6e-15)))
		tmp = x + cos(y);
	else
		tmp = 1.0 + (x + (y * ((y * ((0.16666666666666666 * (y * z)) - 0.5)) - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -12500000000000.0], N[Not[LessEqual[y, 3.6e-15]], $MachinePrecision]], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x + N[(y * N[(N[(y * N[(N[(0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e13 or 3.6000000000000001e-15 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 59.9%

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

      1. +-commutative 59.9%

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

   5. Simplified 59.9%

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

   if -1.25e13 < y < 3.6000000000000001e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12500000000000 \lor \neg \left(y \leq 3.6 \cdot 10^{-15}\right):\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) - 0.5\right) - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.0% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+16} \lor \neg \left(y \leq 1.6 \cdot 10^{+31}\right):\\ \;\;\;\;x \cdot \frac{x + \frac{-1}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.1e+16) (not (<= y 1.6e+31)))
   (* x (/ (+ x (/ -1.0 x)) (+ x -1.0)))
   (+ x (- 1.0 (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.1e+16) || !(y <= 1.6e+31)) {
		tmp = x * ((x + (-1.0 / x)) / (x + -1.0));
	} else {
		tmp = x + (1.0 - (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 <= (-4.1d+16)) .or. (.not. (y <= 1.6d+31))) then
        tmp = x * ((x + ((-1.0d0) / x)) / (x + (-1.0d0)))
    else
        tmp = x + (1.0d0 - (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.1e+16) || !(y <= 1.6e+31)) {
		tmp = x * ((x + (-1.0 / x)) / (x + -1.0));
	} else {
		tmp = x + (1.0 - (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.1e+16) or not (y <= 1.6e+31):
		tmp = x * ((x + (-1.0 / x)) / (x + -1.0))
	else:
		tmp = x + (1.0 - (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.1e+16) || !(y <= 1.6e+31))
		tmp = Float64(x * Float64(Float64(x + Float64(-1.0 / x)) / Float64(x + -1.0)));
	else
		tmp = Float64(x + Float64(1.0 - Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.1e+16) || ~((y <= 1.6e+31)))
		tmp = x * ((x + (-1.0 / x)) / (x + -1.0));
	else
		tmp = x + (1.0 - (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.1e+16], N[Not[LessEqual[y, 1.6e+31]], $MachinePrecision]], N[(x * N[(N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1e16 or 1.6e31 < y

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. add-cube-cbrt 99.2%

         \[\leadsto \left(-\color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \sqrt[3]{z \cdot \sin y}}\right) + \left(x + \cos y\right) \]

      4. distribute-rgt-neg-in 99.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \left(-\sqrt[3]{z \cdot \sin y}\right)} + \left(x + \cos y\right) \]

      5. fma-define 99.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

      6. pow2 99.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right) \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

   5. Taylor expanded in x around inf 88.3%

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

      1. fma-define 88.3%

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

      2. associate-/l* 88.2%

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

   7. Simplified 88.2%

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

      1. distribute-rgt-in 88.2%

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

      2. *-un-lft-identity 88.2%

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

      3. flip-+ 51.3%

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

      4. pow2 51.3%

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

   9. Applied egg-rr 51.3%

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

   11. Simplified 39.6%

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

   12. Taylor expanded in y around 0 22.7%

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

       1. associate-/l* 43.7%

          \[\leadsto \color{blue}{x \cdot \frac{x - \frac{1}{x}}{x - 1}} \]

       2. sub-neg 43.7%

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

       3. metadata-eval 43.7%

          \[\leadsto x \cdot \frac{x - \frac{1}{x}}{x + \color{blue}{-1}} \]

   14. Simplified 43.7%

       \[\leadsto \color{blue}{x \cdot \frac{x - \frac{1}{x}}{x + -1}} \]

   if -4.1e16 < y < 1.6e31

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 94.3%

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

      1. associate-+r+ 94.3%

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

      2. +-commutative 94.3%

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

      3. associate-+l+ 94.3%

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

      4. mul-1-neg 94.3%

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

      5. unsub-neg 94.3%

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

   5. Simplified 94.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+16} \lor \neg \left(y \leq 1.6 \cdot 10^{+31}\right):\\ \;\;\;\;x \cdot \frac{x + \frac{-1}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+16} \lor \neg \left(y \leq 1.35 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot \frac{x + \frac{-1}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot -0.5 - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6e+16) (not (<= y 1.35e+32)))
   (* x (/ (+ x (/ -1.0 x)) (+ x -1.0)))
   (+ 1.0 (+ x (* y (- (* y -0.5) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6e+16) || !(y <= 1.35e+32)) {
		tmp = x * ((x + (-1.0 / x)) / (x + -1.0));
	} else {
		tmp = 1.0 + (x + (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 <= (-6d+16)) .or. (.not. (y <= 1.35d+32))) then
        tmp = x * ((x + ((-1.0d0) / x)) / (x + (-1.0d0)))
    else
        tmp = 1.0d0 + (x + (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 <= -6e+16) || !(y <= 1.35e+32)) {
		tmp = x * ((x + (-1.0 / x)) / (x + -1.0));
	} else {
		tmp = 1.0 + (x + (y * ((y * -0.5) - z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6e+16) or not (y <= 1.35e+32):
		tmp = x * ((x + (-1.0 / x)) / (x + -1.0))
	else:
		tmp = 1.0 + (x + (y * ((y * -0.5) - z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6e+16) || !(y <= 1.35e+32))
		tmp = Float64(x * Float64(Float64(x + Float64(-1.0 / x)) / Float64(x + -1.0)));
	else
		tmp = Float64(1.0 + Float64(x + 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 <= -6e+16) || ~((y <= 1.35e+32)))
		tmp = x * ((x + (-1.0 / x)) / (x + -1.0));
	else
		tmp = 1.0 + (x + (y * ((y * -0.5) - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6e+16], N[Not[LessEqual[y, 1.35e+32]], $MachinePrecision]], N[(x * N[(N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x + N[(y * N[(N[(y * -0.5), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6e16 or 1.35000000000000006e32 < y

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. add-cube-cbrt 99.2%

         \[\leadsto \left(-\color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \sqrt[3]{z \cdot \sin y}}\right) + \left(x + \cos y\right) \]

      4. distribute-rgt-neg-in 99.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \left(-\sqrt[3]{z \cdot \sin y}\right)} + \left(x + \cos y\right) \]

      5. fma-define 99.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

      6. pow2 99.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right) \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

   5. Taylor expanded in x around inf 88.2%

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

      1. fma-define 88.2%

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

      2. associate-/l* 88.1%

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

   7. Simplified 88.1%

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

      1. distribute-rgt-in 88.2%

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

      2. *-un-lft-identity 88.2%

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

      3. flip-+ 50.9%

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

      4. pow2 50.9%

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

   9. Applied egg-rr 50.9%

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

   11. Simplified 39.1%

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

   12. Taylor expanded in y around 0 22.8%

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

       1. associate-/l* 44.0%

          \[\leadsto \color{blue}{x \cdot \frac{x - \frac{1}{x}}{x - 1}} \]

       2. sub-neg 44.0%

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

       3. metadata-eval 44.0%

          \[\leadsto x \cdot \frac{x - \frac{1}{x}}{x + \color{blue}{-1}} \]

   14. Simplified 44.0%

       \[\leadsto \color{blue}{x \cdot \frac{x - \frac{1}{x}}{x + -1}} \]

   if -6e16 < y < 1.35000000000000006e32

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 93.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+16} \lor \neg \left(y \leq 1.35 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot \frac{x + \frac{-1}{x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot -0.5 - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+16} \lor \neg \left(y \leq 1.6 \cdot 10^{+31}\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.6e+16) (not (<= y 1.6e+31)))
   (+ x 1.0)
   (+ x (- 1.0 (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.6e+16) || !(y <= 1.6e+31)) {
		tmp = x + 1.0;
	} else {
		tmp = x + (1.0 - (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 <= (-3.6d+16)) .or. (.not. (y <= 1.6d+31))) then
        tmp = x + 1.0d0
    else
        tmp = x + (1.0d0 - (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.6e+16) || !(y <= 1.6e+31)) {
		tmp = x + 1.0;
	} else {
		tmp = x + (1.0 - (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.6e+16) or not (y <= 1.6e+31):
		tmp = x + 1.0
	else:
		tmp = x + (1.0 - (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.6e+16) || !(y <= 1.6e+31))
		tmp = Float64(x + 1.0);
	else
		tmp = Float64(x + Float64(1.0 - Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.6e+16) || ~((y <= 1.6e+31)))
		tmp = x + 1.0;
	else
		tmp = x + (1.0 - (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.6e+16], N[Not[LessEqual[y, 1.6e+31]], $MachinePrecision]], N[(x + 1.0), $MachinePrecision], N[(x + N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6e16 or 1.6e31 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 43.7%

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

      1. +-commutative 43.7%

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

   5. Simplified 43.7%

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

   if -3.6e16 < y < 1.6e31

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 94.3%

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

      1. associate-+r+ 94.3%

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

      2. +-commutative 94.3%

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

      3. associate-+l+ 94.3%

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

      4. mul-1-neg 94.3%

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

      5. unsub-neg 94.3%

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

   5. Simplified 94.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+16} \lor \neg \left(y \leq 1.6 \cdot 10^{+31}\right):\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.2% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.9e-8) x (if (<= x 1.0) 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.9e-8) {
		tmp = x;
	} else if (x <= 1.0) {
		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 <= (-4.9d-8)) then
        tmp = x
    else if (x <= 1.0d0) 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 <= -4.9e-8) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.9e-8:
		tmp = x
	elif x <= 1.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.9e-8)
		tmp = x;
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.9e-8)
		tmp = x;
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.9e-8], x, If[LessEqual[x, 1.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9000000000000002e-8 or 1 < 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 78.4%

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

   if -4.9000000000000002e-8 < x < 1

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. add-cube-cbrt 99.1%

         \[\leadsto \left(-\color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \sqrt[3]{z \cdot \sin y}}\right) + \left(x + \cos y\right) \]

      4. distribute-rgt-neg-in 99.1%

         \[\leadsto \color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \left(-\sqrt[3]{z \cdot \sin y}\right)} + \left(x + \cos y\right) \]

      5. fma-define 99.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

      6. pow2 99.1%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right) \]

   4. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

   5. Taylor expanded in x around inf 79.6%

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

      1. fma-define 79.6%

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

      2. associate-/l* 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in y around 0 37.4%

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

   9. Taylor expanded in x around 0 37.5%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.0% 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 60.4%

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

   1. +-commutative 60.4%

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

5. Simplified 60.4%

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

6. Final simplification 60.4%

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

Alternative 11: 21.5% 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. add-cube-cbrt 99.4%

      \[\leadsto \left(-\color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \sqrt[3]{z \cdot \sin y}}\right) + \left(x + \cos y\right) \]

   4. distribute-rgt-neg-in 99.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}\right) \cdot \left(-\sqrt[3]{z \cdot \sin y}\right)} + \left(x + \cos y\right) \]

   5. fma-define 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z \cdot \sin y}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

   6. pow2 99.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right) \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{z \cdot \sin y}\right)}^{2}, -\sqrt[3]{z \cdot \sin y}, x + \cos y\right)} \]

5. Taylor expanded in x around inf 90.9%

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

   1. fma-define 90.9%

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

   2. associate-/l* 90.9%

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

7. Simplified 90.9%

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

8. Taylor expanded in y around 0 60.4%

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

9. Taylor expanded in x around 0 18.2%

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

10. Final simplification 18.2%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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))))