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

Percentage Accurate: 99.9% → 99.9%

Time: 7.0s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + \cos y\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -5000000000000 \lor \neg \left(t\_1 \leq 0.9998\right):\\ \;\;\;\;\left(x + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos y + x}{z} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- (+ x (cos y)) t_0)))
   (if (or (<= t_1 -5000000000000.0) (not (<= t_1 0.9998)))
     (- (+ x 1.0) t_0)
     (* (/ (+ (cos y) x) z) z))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = (x + cos(y)) - t_0;
	double tmp;
	if ((t_1 <= -5000000000000.0) || !(t_1 <= 0.9998)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = ((cos(y) + x) / z) * 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) :: t_1
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = (x + cos(y)) - t_0
    if ((t_1 <= (-5000000000000.0d0)) .or. (.not. (t_1 <= 0.9998d0))) then
        tmp = (x + 1.0d0) - t_0
    else
        tmp = ((cos(y) + x) / z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double t_1 = (x + Math.cos(y)) - t_0;
	double tmp;
	if ((t_1 <= -5000000000000.0) || !(t_1 <= 0.9998)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = ((Math.cos(y) + x) / z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = (x + math.cos(y)) - t_0
	tmp = 0
	if (t_1 <= -5000000000000.0) or not (t_1 <= 0.9998):
		tmp = (x + 1.0) - t_0
	else:
		tmp = ((math.cos(y) + x) / z) * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(Float64(x + cos(y)) - t_0)
	tmp = 0.0
	if ((t_1 <= -5000000000000.0) || !(t_1 <= 0.9998))
		tmp = Float64(Float64(x + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(cos(y) + x) / z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = (x + cos(y)) - t_0;
	tmp = 0.0;
	if ((t_1 <= -5000000000000.0) || ~((t_1 <= 0.9998)))
		tmp = (x + 1.0) - t_0;
	else
		tmp = ((cos(y) + x) / z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5000000000000.0], N[Not[LessEqual[t$95$1, 0.9998]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -5e12 or 0.99980000000000002 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.9%

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

   if -5e12 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.99980000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 3.6%

      \[\leadsto \frac{x}{z} \cdot z \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{x + \cos y}{z} \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 99.8%

       \[\leadsto \frac{\cos y + x}{z} \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -5000000000000 \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 0.9998\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos y + x}{z} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+43} \lor \neg \left(t\_2 \leq 0.6\right):\\ \;\;\;\;\left(x + 1\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))) (t_1 (* z (sin y))) (t_2 (- t_0 t_1)))
   (if (or (<= t_2 -4e+43) (not (<= t_2 0.6)))
     (- (+ x 1.0) t_1)
     (- t_0 (* z y)))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double t_1 = z * sin(y);
	double t_2 = t_0 - t_1;
	double tmp;
	if ((t_2 <= -4e+43) || !(t_2 <= 0.6)) {
		tmp = (x + 1.0) - t_1;
	} else {
		tmp = t_0 - (z * 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x + cos(y)
    t_1 = z * sin(y)
    t_2 = t_0 - t_1
    if ((t_2 <= (-4d+43)) .or. (.not. (t_2 <= 0.6d0))) then
        tmp = (x + 1.0d0) - t_1
    else
        tmp = t_0 - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + Math.cos(y);
	double t_1 = z * Math.sin(y);
	double t_2 = t_0 - t_1;
	double tmp;
	if ((t_2 <= -4e+43) || !(t_2 <= 0.6)) {
		tmp = (x + 1.0) - t_1;
	} else {
		tmp = t_0 - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + math.cos(y)
	t_1 = z * math.sin(y)
	t_2 = t_0 - t_1
	tmp = 0
	if (t_2 <= -4e+43) or not (t_2 <= 0.6):
		tmp = (x + 1.0) - t_1
	else:
		tmp = t_0 - (z * y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	t_1 = Float64(z * sin(y))
	t_2 = Float64(t_0 - t_1)
	tmp = 0.0
	if ((t_2 <= -4e+43) || !(t_2 <= 0.6))
		tmp = Float64(Float64(x + 1.0) - t_1);
	else
		tmp = Float64(t_0 - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + cos(y);
	t_1 = z * sin(y);
	t_2 = t_0 - t_1;
	tmp = 0.0;
	if ((t_2 <= -4e+43) || ~((t_2 <= 0.6)))
		tmp = (x + 1.0) - t_1;
	else
		tmp = t_0 - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -4e+43], N[Not[LessEqual[t$95$2, 0.6]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - N[(z * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -4.00000000000000006e43 or 0.599999999999999978 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 96.5%

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

   if -4.00000000000000006e43 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.599999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -4 \cdot 10^{+43} \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 0.6\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + \cos y\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+29} \lor \neg \left(t\_1 \leq 0.6\right):\\ \;\;\;\;\left(x + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- (+ x (cos y)) t_0)))
   (if (or (<= t_1 -1e+29) (not (<= t_1 0.6)))
     (- (+ x 1.0) t_0)
     (- (cos y) (* z y)))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = (x + cos(y)) - t_0;
	double tmp;
	if ((t_1 <= -1e+29) || !(t_1 <= 0.6)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = cos(y) - (z * 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = (x + cos(y)) - t_0
    if ((t_1 <= (-1d+29)) .or. (.not. (t_1 <= 0.6d0))) then
        tmp = (x + 1.0d0) - t_0
    else
        tmp = cos(y) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double t_1 = (x + Math.cos(y)) - t_0;
	double tmp;
	if ((t_1 <= -1e+29) || !(t_1 <= 0.6)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = Math.cos(y) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = (x + math.cos(y)) - t_0
	tmp = 0
	if (t_1 <= -1e+29) or not (t_1 <= 0.6):
		tmp = (x + 1.0) - t_0
	else:
		tmp = math.cos(y) - (z * y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(Float64(x + cos(y)) - t_0)
	tmp = 0.0
	if ((t_1 <= -1e+29) || !(t_1 <= 0.6))
		tmp = Float64(Float64(x + 1.0) - t_0);
	else
		tmp = Float64(cos(y) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = (x + cos(y)) - t_0;
	tmp = 0.0;
	if ((t_1 <= -1e+29) || ~((t_1 <= 0.6)))
		tmp = (x + 1.0) - t_0;
	else
		tmp = cos(y) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+29], N[Not[LessEqual[t$95$1, 0.6]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -9.99999999999999914e28 or 0.599999999999999978 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 96.5%

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

   if -9.99999999999999914e28 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.599999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-cos.f64 69.0

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

   8. Applied rewrites 69.0%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -1 \cdot 10^{+29} \lor \neg \left(\left(x + \cos y\right) - z \cdot \sin y \leq 0.6\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-14} \lor \neg \left(x \leq 6.2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(x + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos y - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (or (<= x -1.85e-14) (not (<= x 6.2e-12)))
     (- (+ x 1.0) t_0)
     (- (cos y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if ((x <= -1.85e-14) || !(x <= 6.2e-12)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = cos(y) - 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 * sin(y)
    if ((x <= (-1.85d-14)) .or. (.not. (x <= 6.2d-12))) then
        tmp = (x + 1.0d0) - t_0
    else
        tmp = cos(y) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double tmp;
	if ((x <= -1.85e-14) || !(x <= 6.2e-12)) {
		tmp = (x + 1.0) - t_0;
	} else {
		tmp = Math.cos(y) - t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	tmp = 0
	if (x <= -1.85e-14) or not (x <= 6.2e-12):
		tmp = (x + 1.0) - t_0
	else:
		tmp = math.cos(y) - t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if ((x <= -1.85e-14) || !(x <= 6.2e-12))
		tmp = Float64(Float64(x + 1.0) - t_0);
	else
		tmp = Float64(cos(y) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	tmp = 0.0;
	if ((x <= -1.85e-14) || ~((x <= 6.2e-12)))
		tmp = (x + 1.0) - t_0;
	else
		tmp = cos(y) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.85e-14], N[Not[LessEqual[x, 6.2e-12]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Cos[y], $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85000000000000001e-14 or 6.2000000000000002e-12 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.5%

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

   if -1.85000000000000001e-14 < x < 6.2000000000000002e-12

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\cos y} - z \cdot \sin y \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-14} \lor \neg \left(x \leq 6.2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \lor \neg \left(z \leq 7.5 \cdot 10^{-14}\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\cos y}{-x} - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.2) (not (<= z 7.5e-14)))
   (- (+ x 1.0) (* z (sin y)))
   (* (- x) (- (/ (cos y) (- x)) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.2) || !(z <= 7.5e-14)) {
		tmp = (x + 1.0) - (z * sin(y));
	} else {
		tmp = -x * ((cos(y) / -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 <= (-1.2d0)) .or. (.not. (z <= 7.5d-14))) then
        tmp = (x + 1.0d0) - (z * sin(y))
    else
        tmp = -x * ((cos(y) / -x) - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.2) || !(z <= 7.5e-14)) {
		tmp = (x + 1.0) - (z * Math.sin(y));
	} else {
		tmp = -x * ((Math.cos(y) / -x) - 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.2) or not (z <= 7.5e-14):
		tmp = (x + 1.0) - (z * math.sin(y))
	else:
		tmp = -x * ((math.cos(y) / -x) - 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.2) || !(z <= 7.5e-14))
		tmp = Float64(Float64(x + 1.0) - Float64(z * sin(y)));
	else
		tmp = Float64(Float64(-x) * Float64(Float64(cos(y) / Float64(-x)) - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.2) || ~((z <= 7.5e-14)))
		tmp = (x + 1.0) - (z * sin(y));
	else
		tmp = -x * ((cos(y) / -x) - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.2], N[Not[LessEqual[z, 7.5e-14]], $MachinePrecision]], N[(N[(x + 1.0), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x) * N[(N[(N[Cos[y], $MachinePrecision] / (-x)), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.19999999999999996 or 7.4999999999999996e-14 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

   if -1.19999999999999996 < z < 7.4999999999999996e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around 0

      \[\leadsto \left(-x\right) \cdot \left(-1 \cdot \frac{\cos y}{x} - 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

       \[\leadsto \left(-x\right) \cdot \left(\frac{\cos y}{-x} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \lor \neg \left(z \leq 7.5 \cdot 10^{-14}\right):\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\cos y}{-x} - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+161} \lor \neg \left(z \leq 9.5 \cdot 10^{+176}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5e+161) (not (<= z 9.5e+176))) (* (- z) (sin y)) (+ 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e+161) || !(z <= 9.5e+176)) {
		tmp = -z * sin(y);
	} else {
		tmp = 1.0 + 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 ((z <= (-5.5d+161)) .or. (.not. (z <= 9.5d+176))) then
        tmp = -z * sin(y)
    else
        tmp = 1.0d0 + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e+161) || !(z <= 9.5e+176)) {
		tmp = -z * Math.sin(y);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.5e+161) or not (z <= 9.5e+176):
		tmp = -z * math.sin(y)
	else:
		tmp = 1.0 + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5e+161) || !(z <= 9.5e+176))
		tmp = Float64(Float64(-z) * sin(y));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.5e+161) || ~((z <= 9.5e+176)))
		tmp = -z * sin(y);
	else
		tmp = 1.0 + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5e+161], N[Not[LessEqual[z, 9.5e+176]], $MachinePrecision]], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5000000000000005e161 or 9.4999999999999995e176 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if -5.5000000000000005e161 < z < 9.4999999999999995e176

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 71.9

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

   5. Applied rewrites 71.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+161} \lor \neg \left(z \leq 9.5 \cdot 10^{+176}\right):\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1500000:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{-1}{x} - 1\right)\\ \mathbf{elif}\;y \leq 38000000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y - 0.5, y, -z\right), y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1500000.0)
   (* (- x) (- (/ -1.0 x) 1.0))
   (if (<= y 38000000000000.0)
     (fma (fma (- (* (* 0.16666666666666666 z) y) 0.5) y (- z)) y (+ 1.0 x))
     (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1500000.0) {
		tmp = -x * ((-1.0 / x) - 1.0);
	} else if (y <= 38000000000000.0) {
		tmp = fma(fma((((0.16666666666666666 * z) * y) - 0.5), y, -z), y, (1.0 + x));
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1500000.0)
		tmp = Float64(Float64(-x) * Float64(Float64(-1.0 / x) - 1.0));
	elseif (y <= 38000000000000.0)
		tmp = fma(fma(Float64(Float64(Float64(0.16666666666666666 * z) * y) - 0.5), y, Float64(-z)), y, Float64(1.0 + x));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1500000.0], N[((-x) * N[(N[(-1.0 / x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 38000000000000.0], N[(N[(N[(N[(N[(0.16666666666666666 * z), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y + (-z)), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e6

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 85.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 39.3%

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

   if -1.5e6 < y < 3.8e13

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 97.0%

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

   if 3.8e13 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 34.2

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

   5. Applied rewrites 34.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1500000:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{-1}{x} - 1\right)\\ \mathbf{elif}\;y \leq 38000000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y - 0.5, y, -z\right), y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -42000 \lor \neg \left(y \leq 1.35 \cdot 10^{+51}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -42000.0) (not (<= y 1.35e+51)))
   (+ 1.0 x)
   (fma (- (* -0.5 y) z) y (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -42000.0) || !(y <= 1.35e+51)) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(((-0.5 * y) - z), y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -42000.0) || !(y <= 1.35e+51))
		tmp = Float64(1.0 + x);
	else
		tmp = fma(Float64(Float64(-0.5 * y) - z), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -42000.0], N[Not[LessEqual[y, 1.35e+51]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[(N[(N[(-0.5 * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -42000 or 1.34999999999999996e51 < 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

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

      1. lower-+.f64 36.4

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

   5. Applied rewrites 36.4%

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

   if -42000 < y < 1.34999999999999996e51

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -42000 \lor \neg \left(y \leq 1.35 \cdot 10^{+51}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -42000:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{-1}{x} - 1\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -42000.0)
   (* (- x) (- (/ -1.0 x) 1.0))
   (if (<= y 1.35e+51) (fma (- (* -0.5 y) z) y (+ 1.0 x)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -42000.0) {
		tmp = -x * ((-1.0 / x) - 1.0);
	} else if (y <= 1.35e+51) {
		tmp = fma(((-0.5 * y) - z), y, (1.0 + x));
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -42000.0)
		tmp = Float64(Float64(-x) * Float64(Float64(-1.0 / x) - 1.0));
	elseif (y <= 1.35e+51)
		tmp = fma(Float64(Float64(-0.5 * y) - z), y, Float64(1.0 + x));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -42000.0], N[((-x) * N[(N[(-1.0 / x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+51], N[(N[(N[(-0.5 * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -42000

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 85.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 38.6%

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

   if -42000 < y < 1.34999999999999996e51

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 94.9

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

   5. Applied rewrites 94.9%

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

   if 1.34999999999999996e51 < 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

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

      1. lower-+.f64 34.1

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

   5. Applied rewrites 34.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -42000:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{-1}{x} - 1\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{+26} \lor \neg \left(y \leq 4.1 \cdot 10^{+52}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.95e+26) (not (<= y 4.1e+52)))
   (+ 1.0 x)
   (fma (- z) y (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.95e+26) || !(y <= 4.1e+52)) {
		tmp = 1.0 + x;
	} else {
		tmp = fma(-z, y, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.95e+26) || !(y <= 4.1e+52))
		tmp = Float64(1.0 + x);
	else
		tmp = fma(Float64(-z), y, Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.95e+26], N[Not[LessEqual[y, 4.1e+52]], $MachinePrecision]], N[(1.0 + x), $MachinePrecision], N[((-z) * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.95000000000000015e26 or 4.1e52 < 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

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

      1. lower-+.f64 36.2

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

   5. Applied rewrites 36.2%

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

   if -2.95000000000000015e26 < y < 4.1e52

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 91.1

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

   5. Applied rewrites 91.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{+26} \lor \neg \left(y \leq 4.1 \cdot 10^{+52}\right):\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.4% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+164}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.2e+164) (* (- y) z) (+ 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.2e+164) {
		tmp = -y * z;
	} else {
		tmp = 1.0 + 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 (z <= (-8.2d+164)) then
        tmp = -y * z
    else
        tmp = 1.0d0 + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.2e+164) {
		tmp = -y * z;
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.2e+164:
		tmp = -y * z
	else:
		tmp = 1.0 + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.2e+164)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8.2e+164)
		tmp = -y * z;
	else
		tmp = 1.0 + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.2e+164], N[((-y) * z), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.20000000000000032e164

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 40.5%

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

   if -8.20000000000000032e164 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 65.0

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

   5. Applied rewrites 65.0%

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

Alternative 13: 61.9% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

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 + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. lower-+.f64 59.3

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

5. Applied rewrites 59.3%

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

Alternative 14: 21.8% accurate, 212.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. lower-+.f64 59.3

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

5. Applied rewrites 59.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.0%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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))))