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

Percentage Accurate: 99.9% → 99.9%

Time: 10.6s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ t_3 := \left(x + 1\right) - t\_1\\ \mathbf{if}\;t\_2 \leq -150000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y)))
        (t_1 (* z (sin y)))
        (t_2 (- t_0 t_1))
        (t_3 (- (+ x 1.0) t_1)))
   (if (<= t_2 -150000.0) t_3 (if (<= t_2 0.995) t_0 t_3))))

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 t_3 = (x + 1.0) - t_1;
	double tmp;
	if (t_2 <= -150000.0) {
		tmp = t_3;
	} else if (t_2 <= 0.995) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = x + cos(y)
    t_1 = z * sin(y)
    t_2 = t_0 - t_1
    t_3 = (x + 1.0d0) - t_1
    if (t_2 <= (-150000.0d0)) then
        tmp = t_3
    else if (t_2 <= 0.995d0) then
        tmp = t_0
    else
        tmp = t_3
    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 t_3 = (x + 1.0) - t_1;
	double tmp;
	if (t_2 <= -150000.0) {
		tmp = t_3;
	} else if (t_2 <= 0.995) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	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
	t_3 = (x + 1.0) - t_1
	tmp = 0
	if t_2 <= -150000.0:
		tmp = t_3
	elif t_2 <= 0.995:
		tmp = t_0
	else:
		tmp = t_3
	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)
	t_3 = Float64(Float64(x + 1.0) - t_1)
	tmp = 0.0
	if (t_2 <= -150000.0)
		tmp = t_3;
	elseif (t_2 <= 0.995)
		tmp = t_0;
	else
		tmp = t_3;
	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;
	t_3 = (x + 1.0) - t_1;
	tmp = 0.0;
	if (t_2 <= -150000.0)
		tmp = t_3;
	elseif (t_2 <= 0.995)
		tmp = t_0;
	else
		tmp = t_3;
	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]}, Block[{t$95$3 = N[(N[(x + 1.0), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -150000.0], t$95$3, If[LessEqual[t$95$2, 0.995], t$95$0, t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1.5e5 or 0.994999999999999996 < (-.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 99.6%

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

   if -1.5e5 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.994999999999999996

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -150000:\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 0.995:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := t\_0 - z \cdot \sin y\\ t_2 := \mathsf{fma}\left(z, -\sin y, x\right)\\ \mathbf{if}\;t\_1 \leq -150000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y)))
        (t_1 (- t_0 (* z (sin y))))
        (t_2 (fma z (- (sin y)) x)))
   (if (<= t_1 -150000.0) t_2 (if (<= t_1 2.0) t_0 t_2))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double t_1 = t_0 - (z * sin(y));
	double t_2 = fma(z, -sin(y), x);
	double tmp;
	if (t_1 <= -150000.0) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	t_1 = Float64(t_0 - Float64(z * sin(y)))
	t_2 = fma(z, Float64(-sin(y)), x)
	tmp = 0.0
	if (t_1 <= -150000.0)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * (-N[Sin[y], $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[t$95$1, -150000.0], t$95$2, If[LessEqual[t$95$1, 2.0], t$95$0, t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1.5e5 or 2 < (-.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

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

      1. +-commutative N/A

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

      2. lower-+.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 86.9

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.9%

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

   12. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(z, -1 \cdot \sin y, x\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.6%

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

   if -1.5e5 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -150000:\\ \;\;\;\;\mathsf{fma}\left(z, -\sin y, x\right)\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 2:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, -\sin y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + 1\right) - t\_0\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-54}:\\ \;\;\;\;\cos y - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- (+ x 1.0) t_0)))
   (if (<= x -1.45e-8) t_1 (if (<= x 6e-54) (- (cos y) t_0) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = (x + 1.0) - t_0;
	double tmp;
	if (x <= -1.45e-8) {
		tmp = t_1;
	} else if (x <= 6e-54) {
		tmp = cos(y) - t_0;
	} else {
		tmp = t_1;
	}
	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 + 1.0d0) - t_0
    if (x <= (-1.45d-8)) then
        tmp = t_1
    else if (x <= 6d-54) then
        tmp = cos(y) - t_0
    else
        tmp = t_1
    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 + 1.0) - t_0;
	double tmp;
	if (x <= -1.45e-8) {
		tmp = t_1;
	} else if (x <= 6e-54) {
		tmp = Math.cos(y) - t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = (x + 1.0) - t_0
	tmp = 0
	if x <= -1.45e-8:
		tmp = t_1
	elif x <= 6e-54:
		tmp = math.cos(y) - t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(Float64(x + 1.0) - t_0)
	tmp = 0.0
	if (x <= -1.45e-8)
		tmp = t_1;
	elseif (x <= 6e-54)
		tmp = Float64(cos(y) - t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = (x + 1.0) - t_0;
	tmp = 0.0;
	if (x <= -1.45e-8)
		tmp = t_1;
	elseif (x <= 6e-54)
		tmp = cos(y) - t_0;
	else
		tmp = t_1;
	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 + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[x, -1.45e-8], t$95$1, If[LessEqual[x, 6e-54], N[(N[Cos[y], $MachinePrecision] - t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4500000000000001e-8 or 6.00000000000000018e-54 < 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 -1.4500000000000001e-8 < x < 6.00000000000000018e-54

   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 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 5: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ \mathbf{if}\;y \leq -1.65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))))
   (if (<= y -1.65)
     t_0
     (if (<= y 1.15e+14)
       (-
        (+ x 1.0)
        (*
         y
         (fma
          (* y y)
          (*
           z
           (fma
            y
            (* y (fma (* y y) -0.0001984126984126984 0.008333333333333333))
            -0.16666666666666666))
          z)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double tmp;
	if (y <= -1.65) {
		tmp = t_0;
	} else if (y <= 1.15e+14) {
		tmp = (x + 1.0) - (y * fma((y * y), (z * fma(y, (y * fma((y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	tmp = 0.0
	if (y <= -1.65)
		tmp = t_0;
	elseif (y <= 1.15e+14)
		tmp = Float64(Float64(x + 1.0) - Float64(y * fma(Float64(y * y), Float64(z * fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), z)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.65], t$95$0, If[LessEqual[y, 1.15e+14], N[(N[(x + 1.0), $MachinePrecision] - N[(y * N[(N[(y * y), $MachinePrecision] * N[(z * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6499999999999999 or 1.15e14 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if -1.6499999999999999 < y < 1.15e14

   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 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right) + z\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right), z\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(z \cdot \mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right)\right), z \cdot -0.16666666666666666\right), z\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right) + z\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right), z\right)} \]

   11. Applied rewrites 99.3%

       \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65:\\ \;\;\;\;x + \cos y\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -34000000:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -34000000.0)
   (+ x 1.0)
   (if (<= y 5.8e+14)
     (-
      (+ x 1.0)
      (*
       y
       (fma
        (* y y)
        (*
         z
         (fma
          y
          (* y (fma (* y y) -0.0001984126984126984 0.008333333333333333))
          -0.16666666666666666))
        z)))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -34000000.0) {
		tmp = x + 1.0;
	} else if (y <= 5.8e+14) {
		tmp = (x + 1.0) - (y * fma((y * y), (z * fma(y, (y * fma((y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), z));
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -34000000.0)
		tmp = Float64(x + 1.0);
	elseif (y <= 5.8e+14)
		tmp = Float64(Float64(x + 1.0) - Float64(y * fma(Float64(y * y), Float64(z * fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), z)));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -34000000.0], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 5.8e+14], N[(N[(x + 1.0), $MachinePrecision] - N[(y * N[(N[(y * y), $MachinePrecision] * N[(z * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e7 or 5.8e14 < 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. +-commutative N/A

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

      2. lower-+.f64 42.1

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

   5. Applied rewrites 42.1%

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

   if -3.4e7 < y < 5.8e14

   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 99.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right) + z\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right), z\right)} \]

   8. Applied rewrites 98.6%

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \left(z \cdot \mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right)\right), z \cdot -0.16666666666666666\right), z\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right)\right) + z\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({y}^{2} \cdot z\right) + \frac{1}{120} \cdot z\right), z\right)} \]

   11. Applied rewrites 98.6%

       \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+45}:\\ \;\;\;\;\left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.7e+14)
   (+ x 1.0)
   (if (<= y 1.05e+45)
     (-
      (+ x 1.0)
      (*
       y
       (fma
        (* y y)
        (* z (fma 0.008333333333333333 (* y y) -0.16666666666666666))
        z)))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+14) {
		tmp = x + 1.0;
	} else if (y <= 1.05e+45) {
		tmp = (x + 1.0) - (y * fma((y * y), (z * fma(0.008333333333333333, (y * y), -0.16666666666666666)), z));
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.7e+14)
		tmp = Float64(x + 1.0);
	elseif (y <= 1.05e+45)
		tmp = Float64(Float64(x + 1.0) - Float64(y * fma(Float64(y * y), Float64(z * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666)), z)));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.7e+14], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 1.05e+45], N[(N[(x + 1.0), $MachinePrecision] - N[(y * N[(N[(y * y), $MachinePrecision] * N[(z * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e14 or 1.04999999999999997e45 < 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. +-commutative N/A

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

      2. lower-+.f64 41.8

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

   5. Applied rewrites 41.8%

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

   if -1.7e14 < y < 1.04999999999999997e45

   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 99.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \left(z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right)\right) + z\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + 1\right) - y \cdot \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right), z\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(x + 1\right) - y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right), z\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(x + 1\right) - y \cdot \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{6} \cdot z + \frac{1}{120} \cdot \left({y}^{2} \cdot z\right), z\right) \]

      6. +-commutative N/A

         \[\leadsto \left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{120} \cdot \left({y}^{2} \cdot z\right) + \frac{-1}{6} \cdot z}, z\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot z} + \frac{-1}{6} \cdot z, z\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{z \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)}, z\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{z \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)}, z\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {y}^{2}, \frac{-1}{6}\right)}, z\right) \]

      11. unpow2 N/A

          \[\leadsto \left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{y \cdot y}, \frac{-1}{6}\right), z\right) \]

      12. lower-*.f64 97.1

          \[\leadsto \left(x + 1\right) - y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{y \cdot y}, -0.16666666666666666\right), z\right) \]

   8. Applied rewrites 97.1%

      \[\leadsto \left(x + 1\right) - \color{blue}{y \cdot \mathsf{fma}\left(y \cdot y, z \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3200000:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;1 + \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, z \cdot 0.16666666666666666, -0.5\right) - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3200000.0)
   (+ x 1.0)
   (if (<= y 1.75e+14)
     (+ 1.0 (fma y (- (* y (fma y (* z 0.16666666666666666) -0.5)) z) x))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3200000.0) {
		tmp = x + 1.0;
	} else if (y <= 1.75e+14) {
		tmp = 1.0 + fma(y, ((y * fma(y, (z * 0.16666666666666666), -0.5)) - z), x);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3200000.0)
		tmp = Float64(x + 1.0);
	elseif (y <= 1.75e+14)
		tmp = Float64(1.0 + fma(y, Float64(Float64(y * fma(y, Float64(z * 0.16666666666666666), -0.5)) - z), x));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3200000.0], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 1.75e+14], N[(1.0 + N[(y * N[(N[(y * N[(y * N[(z * 0.16666666666666666), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2e6 or 1.75e14 < 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. +-commutative N/A

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

      2. lower-+.f64 42.1

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

   5. Applied rewrites 42.1%

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

   if -3.2e6 < y < 1.75e14

   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. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 9: 69.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+41}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+114}:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e+41)
   (+ x 1.0)
   (if (<= y 4.7e+114) (- x (fma y z -1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e+41) {
		tmp = x + 1.0;
	} else if (y <= 4.7e+114) {
		tmp = x - fma(y, z, -1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e+41)
		tmp = Float64(x + 1.0);
	elseif (y <= 4.7e+114)
		tmp = Float64(x - fma(y, z, -1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e+41], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 4.7e+114], N[(x - N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000022e41 or 4.7000000000000001e114 < 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. +-commutative N/A

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

      2. lower-+.f64 43.2

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

   5. Applied rewrites 43.2%

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

   if -5.00000000000000022e41 < y < 4.7000000000000001e114

   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. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 87.9

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

   5. Applied rewrites 87.9%

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

Alternative 10: 61.3% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+169}:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4e+169) (- (* y z)) (+ x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4e+169) {
		tmp = -(y * z);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5.4d+169)) then
        tmp = -(y * z)
    else
        tmp = x + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4e+169) {
		tmp = -(y * z);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.4e+169:
		tmp = -(y * z)
	else:
		tmp = x + 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4e+169)
		tmp = Float64(-Float64(y * z));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.4e+169)
		tmp = -(y * z);
	else
		tmp = x + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4e+169], (-N[(y * z), $MachinePrecision]), N[(x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.39999999999999981e169

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   6. Taylor expanded in z around -inf

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

   8. Applied rewrites 49.8%

      \[\leadsto -y \cdot \mathsf{fma}\left(y, z \cdot \left(y \cdot -0.16666666666666666\right), z\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{neg}\left(y \cdot z\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 50.7%

       \[\leadsto -y \cdot z \]

   if -5.39999999999999981e169 < 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. +-commutative N/A

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

      2. lower-+.f64 66.2

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

   5. Applied rewrites 66.2%

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

Alternative 11: 61.2% accurate, 53.0× 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

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

   1. +-commutative N/A

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

   2. lower-+.f64 61.8

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

5. Applied rewrites 61.8%

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

Alternative 12: 21.1% 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. +-commutative N/A

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

   2. lower-+.f64 61.8

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

5. Applied rewrites 61.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 23.9%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024238 
(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))))