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

Percentage Accurate: 99.9% → 99.9%

Time: 7.9s

Alternatives: 15

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ t_3 := \left(1 + x\right) - t\_1\\ \mathbf{if}\;t\_2 \leq -5000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.99:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (- (+ 1.0 x) t_1)))
   (if (<= t_2 -5000.0) t_3 (if (<= t_2 0.99) (/ 1.0 (/ 1.0 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 = (1.0 + x) - t_1;
	double tmp;
	if (t_2 <= -5000.0) {
		tmp = t_3;
	} else if (t_2 <= 0.99) {
		tmp = 1.0 / (1.0 / 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 = (1.0d0 + x) - t_1
    if (t_2 <= (-5000.0d0)) then
        tmp = t_3
    else if (t_2 <= 0.99d0) then
        tmp = 1.0d0 / (1.0d0 / 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 = (1.0 + x) - t_1;
	double tmp;
	if (t_2 <= -5000.0) {
		tmp = t_3;
	} else if (t_2 <= 0.99) {
		tmp = 1.0 / (1.0 / 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 = (1.0 + x) - t_1
	tmp = 0
	if t_2 <= -5000.0:
		tmp = t_3
	elif t_2 <= 0.99:
		tmp = 1.0 / (1.0 / 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(1.0 + x) - t_1)
	tmp = 0.0
	if (t_2 <= -5000.0)
		tmp = t_3;
	elseif (t_2 <= 0.99)
		tmp = Float64(1.0 / Float64(1.0 / 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 = (1.0 + x) - t_1;
	tmp = 0.0;
	if (t_2 <= -5000.0)
		tmp = t_3;
	elseif (t_2 <= 0.99)
		tmp = 1.0 / (1.0 / 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[(1.0 + x), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -5000.0], t$95$3, If[LessEqual[t$95$2, 0.99], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -5e3 or 0.98999999999999999 < (-.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 99.8%

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

   if -5e3 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.98999999999999999

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x + \cos y}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x + \cos y}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\cos y + x}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\cos y + x}}} \]

      4. lower-cos.f64 97.5

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\cos y} + x}} \]

   7. Applied rewrites 97.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -5000:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 0.99:\\ \;\;\;\;\frac{1}{\frac{1}{x + \cos y}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + \cos y\right) - t\_0\\ t_2 := \left(1 + x\right) - t\_0\\ \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.98:\\ \;\;\;\;\cos y - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y)))
        (t_1 (- (+ x (cos y)) t_0))
        (t_2 (- (+ 1.0 x) t_0)))
   (if (<= t_1 -500.0) t_2 (if (<= t_1 0.98) (- (cos y) (* z y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = (x + cos(y)) - t_0;
	double t_2 = (1.0 + x) - t_0;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 0.98) {
		tmp = cos(y) - (z * y);
	} else {
		tmp = t_2;
	}
	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 = z * sin(y)
    t_1 = (x + cos(y)) - t_0
    t_2 = (1.0d0 + x) - t_0
    if (t_1 <= (-500.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.98d0) then
        tmp = cos(y) - (z * y)
    else
        tmp = t_2
    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 t_2 = (1.0 + x) - t_0;
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_2;
	} else if (t_1 <= 0.98) {
		tmp = Math.cos(y) - (z * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = (x + math.cos(y)) - t_0
	t_2 = (1.0 + x) - t_0
	tmp = 0
	if t_1 <= -500.0:
		tmp = t_2
	elif t_1 <= 0.98:
		tmp = math.cos(y) - (z * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(Float64(x + cos(y)) - t_0)
	t_2 = Float64(Float64(1.0 + x) - t_0)
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 0.98)
		tmp = Float64(cos(y) - Float64(z * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = (x + cos(y)) - t_0;
	t_2 = (1.0 + x) - t_0;
	tmp = 0.0;
	if (t_1 <= -500.0)
		tmp = t_2;
	elseif (t_1 <= 0.98)
		tmp = cos(y) - (z * y);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(1.0 + x), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -500.0], t$95$2, If[LessEqual[t$95$1, 0.98], N[(N[Cos[y], $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -500 or 0.97999999999999998 < (-.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 99.3%

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

   if -500 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.97999999999999998

   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 46.4

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right) - z \cdot y \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)}\right) - z \cdot y \]

      2. *-commutative N/A

         \[\leadsto \left(x + \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right)\right) - z \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, {y}^{2}, 1\right)}\right) - z \cdot y \]

      4. sub-neg N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {y}^{2}, 1\right)\right) - z \cdot y \]

      5. *-commutative N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}, {y}^{2}, 1\right)\right) - z \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, {y}^{2}, \frac{-1}{2}\right)}, {y}^{2}, 1\right)\right) - z \cdot y \]

      8. +-commutative N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {y}^{2}, \frac{1}{24}\right)}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      10. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      11. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      12. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      14. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), y \cdot y, \frac{-1}{2}\right), \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

      15. lower-*.f64 1.7

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

   8. Applied rewrites 1.7%

      \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right)}\right) - z \cdot y \]

   9. Taylor expanded in x around 0

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

       1. lower-cos.f64 44.6

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

   11. Applied rewrites 44.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -500:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 0.98:\\ \;\;\;\;\cos y - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 91.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + x\right) - z \cdot \sin y\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{-207}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-106}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ 1.0 x) (* z (sin y)))))
   (if (<= z -6.2e-207)
     t_0
     (if (<= z 1.25e-106) (- (+ x (cos y)) (* z y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 + x) - (z * sin(y));
	double tmp;
	if (z <= -6.2e-207) {
		tmp = t_0;
	} else if (z <= 1.25e-106) {
		tmp = (x + cos(y)) - (z * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + x) - (z * sin(y))
    if (z <= (-6.2d-207)) then
        tmp = t_0
    else if (z <= 1.25d-106) then
        tmp = (x + cos(y)) - (z * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (1.0 + x) - (z * Math.sin(y));
	double tmp;
	if (z <= -6.2e-207) {
		tmp = t_0;
	} else if (z <= 1.25e-106) {
		tmp = (x + Math.cos(y)) - (z * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 + x) - (z * math.sin(y))
	tmp = 0
	if z <= -6.2e-207:
		tmp = t_0
	elif z <= 1.25e-106:
		tmp = (x + math.cos(y)) - (z * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 + x) - Float64(z * sin(y)))
	tmp = 0.0
	if (z <= -6.2e-207)
		tmp = t_0;
	elseif (z <= 1.25e-106)
		tmp = Float64(Float64(x + cos(y)) - Float64(z * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 + x) - (z * sin(y));
	tmp = 0.0;
	if (z <= -6.2e-207)
		tmp = t_0;
	elseif (z <= 1.25e-106)
		tmp = (x + cos(y)) - (z * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.2e-207], t$95$0, If[LessEqual[z, 1.25e-106], N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2000000000000003e-207 or 1.24999999999999996e-106 < 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

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

   5. Applied rewrites 94.5%

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

   if -6.2000000000000003e-207 < z < 1.24999999999999996e-106

   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 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-207}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-106}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;y \leq -0.84:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 104:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))))
   (if (<= y -0.84)
     t_0
     (if (<= y 104.0)
       (-
        (+
         (fma
          (fma
           (fma -0.001388888888888889 (* y y) 0.041666666666666664)
           (* y y)
           -0.5)
          (* y y)
          1.0)
         x)
        (* z y))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double tmp;
	if (y <= -0.84) {
		tmp = t_0;
	} else if (y <= 104.0) {
		tmp = (fma(fma(fma(-0.001388888888888889, (y * y), 0.041666666666666664), (y * y), -0.5), (y * y), 1.0) + x) - (z * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * sin(y))
	tmp = 0.0
	if (y <= -0.84)
		tmp = t_0;
	elseif (y <= 104.0)
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(y * y), 0.041666666666666664), Float64(y * y), -0.5), Float64(y * y), 1.0) + x) - Float64(z * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.84], t$95$0, If[LessEqual[y, 104.0], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(y * y), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.839999999999999969 or 104 < y

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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 47.7

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

   5. Applied rewrites 47.7%

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

   if -0.839999999999999969 < y < 104

   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right) - z \cdot y \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)}\right) - z \cdot y \]

      2. *-commutative N/A

         \[\leadsto \left(x + \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right)\right) - z \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, {y}^{2}, 1\right)}\right) - z \cdot y \]

      4. sub-neg N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {y}^{2}, 1\right)\right) - z \cdot y \]

      5. *-commutative N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}, {y}^{2}, 1\right)\right) - z \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, {y}^{2}, \frac{-1}{2}\right)}, {y}^{2}, 1\right)\right) - z \cdot y \]

      8. +-commutative N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {y}^{2}, \frac{1}{24}\right)}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      10. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      11. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      12. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      14. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), y \cdot y, \frac{-1}{2}\right), \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

      15. lower-*.f64 100.0

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

   8. Applied rewrites 100.0%

      \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right)}\right) - z \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.84:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{elif}\;y \leq 104:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -150000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 325000000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -150000.0)
   (+ 1.0 x)
   (if (<= y 325000000000.0)
     (-
      (+
       (fma
        (fma
         (fma -0.001388888888888889 (* y y) 0.041666666666666664)
         (* y y)
         -0.5)
        (* y y)
        1.0)
       x)
      (* z y))
     (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -150000.0) {
		tmp = 1.0 + x;
	} else if (y <= 325000000000.0) {
		tmp = (fma(fma(fma(-0.001388888888888889, (y * y), 0.041666666666666664), (y * y), -0.5), (y * y), 1.0) + x) - (z * y);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -150000.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 325000000000.0)
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(y * y), 0.041666666666666664), Float64(y * y), -0.5), Float64(y * y), 1.0) + x) - Float64(z * y));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -150000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 325000000000.0], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(y * y), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5e5 or 3.25e11 < 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 35.4

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

   5. Applied rewrites 35.4%

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

   if -1.5e5 < y < 3.25e11

   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 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right) - z \cdot y \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)}\right) - z \cdot y \]

      2. *-commutative N/A

         \[\leadsto \left(x + \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right)\right) - z \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, {y}^{2}, 1\right)}\right) - z \cdot y \]

      4. sub-neg N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {y}^{2}, 1\right)\right) - z \cdot y \]

      5. *-commutative N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}, {y}^{2}, 1\right)\right) - z \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, {y}^{2}, \frac{-1}{2}\right)}, {y}^{2}, 1\right)\right) - z \cdot y \]

      8. +-commutative N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {y}^{2}, \frac{1}{24}\right)}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      10. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      11. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      12. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      14. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), y \cdot y, \frac{-1}{2}\right), \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

      15. lower-*.f64 96.8

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

   8. Applied rewrites 96.8%

      \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right)}\right) - z \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -150000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 325000000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -45000000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 95000000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -45000000000.0)
   (+ 1.0 x)
   (if (<= y 95000000000.0)
     (-
      (+ (fma (fma 0.041666666666666664 (* y y) -0.5) (* y y) 1.0) x)
      (* z y))
     (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -45000000000.0) {
		tmp = 1.0 + x;
	} else if (y <= 95000000000.0) {
		tmp = (fma(fma(0.041666666666666664, (y * y), -0.5), (y * y), 1.0) + x) - (z * y);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -45000000000.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 95000000000.0)
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(y * y), -0.5), Float64(y * y), 1.0) + x) - Float64(z * y));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -45000000000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 95000000000.0], N[(N[(N[(N[(0.041666666666666664 * N[(y * y), $MachinePrecision] + -0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e10 or 9.5e10 < 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 35.1

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

   5. Applied rewrites 35.1%

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

   if -4.5e10 < y < 9.5e10

   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 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)}\right) - z \cdot y \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right) + 1\right)}\right) - z \cdot y \]

      2. *-commutative N/A

         \[\leadsto \left(x + \left(\color{blue}{\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right)\right) - z \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)}\right) - z \cdot y \]

      4. sub-neg N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {y}^{2}, 1\right)\right) - z \cdot y \]

      5. metadata-eval N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\frac{1}{24} \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}, {y}^{2}, 1\right)\right) - z \cdot y \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {y}^{2}, \frac{-1}{2}\right)}, {y}^{2}, 1\right)\right) - z \cdot y \]

      7. unpow2 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      8. lower-*.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      9. unpow2 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, y \cdot y, \frac{-1}{2}\right), \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

      10. lower-*.f64 97.4

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

   8. Applied rewrites 97.4%

      \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right)}\right) - z \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -45000000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 95000000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot y, -0.5\right), y \cdot y, 1\right) + x\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -66000000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 325000000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -66000000000.0)
   (+ 1.0 x)
   (if (<= y 325000000000.0) (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 <= -66000000000.0) {
		tmp = 1.0 + x;
	} else if (y <= 325000000000.0) {
		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 <= -66000000000.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 325000000000.0)
		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, -66000000000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 325000000000.0], 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 2 regimes

2. if y < -6.6e10 or 3.25e11 < 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 35.4

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

   5. Applied rewrites 35.4%

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

   if -6.6e10 < y < 3.25e11

   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 96.6

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

   5. Applied rewrites 96.6%

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

Alternative 10: 69.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+102}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.7e+102)
   (+ 1.0 x)
   (if (<= y 2.3e+125) (fma (- z) y (+ 1.0 x)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.7e+102) {
		tmp = 1.0 + x;
	} else if (y <= 2.3e+125) {
		tmp = fma(-z, y, (1.0 + x));
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.7e+102)
		tmp = Float64(1.0 + x);
	elseif (y <= 2.3e+125)
		tmp = fma(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, -4.7e+102], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 2.3e+125], N[((-z) * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7000000000000003e102 or 2.30000000000000013e125 < 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 36.5

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

   5. Applied rewrites 36.5%

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

   if -4.7000000000000003e102 < y < 2.30000000000000013e125

   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. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.4%

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

Alternative 11: 69.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+102}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+125}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.7e+102)
   (+ 1.0 x)
   (if (<= y 2.3e+125) (- x (fma z y -1.0)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.7e+102) {
		tmp = 1.0 + x;
	} else if (y <= 2.3e+125) {
		tmp = x - fma(z, y, -1.0);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.7e+102)
		tmp = Float64(1.0 + x);
	elseif (y <= 2.3e+125)
		tmp = Float64(x - fma(z, y, -1.0));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.7e+102], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 2.3e+125], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7000000000000003e102 or 2.30000000000000013e125 < 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 36.5

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

   5. Applied rewrites 36.5%

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

   if -4.7000000000000003e102 < y < 2.30000000000000013e125

   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. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 84.4

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

   5. Applied rewrites 84.4%

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

Alternative 12: 64.9% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-162}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+17}:\\ \;\;\;\;1 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.65e-162)
   (+ 1.0 x)
   (if (<= x 2.25e+17) (- 1.0 (* z y)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.65e-162) {
		tmp = 1.0 + x;
	} else if (x <= 2.25e+17) {
		tmp = 1.0 - (z * 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 (x <= (-1.65d-162)) then
        tmp = 1.0d0 + x
    else if (x <= 2.25d+17) then
        tmp = 1.0d0 - (z * 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 (x <= -1.65e-162) {
		tmp = 1.0 + x;
	} else if (x <= 2.25e+17) {
		tmp = 1.0 - (z * y);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.65e-162:
		tmp = 1.0 + x
	elif x <= 2.25e+17:
		tmp = 1.0 - (z * y)
	else:
		tmp = 1.0 + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.65e-162)
		tmp = Float64(1.0 + x);
	elseif (x <= 2.25e+17)
		tmp = Float64(1.0 - Float64(z * y));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.65e-162)
		tmp = 1.0 + x;
	elseif (x <= 2.25e+17)
		tmp = 1.0 - (z * y);
	else
		tmp = 1.0 + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.65e-162], N[(1.0 + x), $MachinePrecision], If[LessEqual[x, 2.25e+17], N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000007e-162 or 2.25e17 < x

   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 72.9

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

   5. Applied rewrites 72.9%

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

   if -1.65000000000000007e-162 < x < 2.25e17

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(x + \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right) - z \cdot y \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + 1\right)}\right) - z \cdot y \]

      2. *-commutative N/A

         \[\leadsto \left(x + \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) \cdot {y}^{2}} + 1\right)\right) - z \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, {y}^{2}, 1\right)}\right) - z \cdot y \]

      4. sub-neg N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {y}^{2}, 1\right)\right) - z \cdot y \]

      5. *-commutative N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} + \color{blue}{\frac{-1}{2}}, {y}^{2}, 1\right)\right) - z \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}, {y}^{2}, \frac{-1}{2}\right)}, {y}^{2}, 1\right)\right) - z \cdot y \]

      8. +-commutative N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {y}^{2}, \frac{1}{24}\right)}, {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      10. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      11. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{y \cdot y}, \frac{1}{24}\right), {y}^{2}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      12. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), \color{blue}{y \cdot y}, \frac{-1}{2}\right), {y}^{2}, 1\right)\right) - z \cdot y \]

      14. unpow2 N/A

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, y \cdot y, \frac{1}{24}\right), y \cdot y, \frac{-1}{2}\right), \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

      15. lower-*.f64 56.1

          \[\leadsto \left(x + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), \color{blue}{y \cdot y}, 1\right)\right) - z \cdot y \]

   8. Applied rewrites 56.1%

      \[\leadsto \left(x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, y \cdot y, 0.041666666666666664\right), y \cdot y, -0.5\right), y \cdot y, 1\right)}\right) - z \cdot y \]

   9. Taylor expanded in x around 0

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

       1. lower-cos.f64 63.0

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

   11. Applied rewrites 63.0%

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

   12. Taylor expanded in y around 0

       \[\leadsto 1 - z \cdot y \]
   13. Step-by-step derivation

   14. Applied rewrites 57.1%

       \[\leadsto 1 - z \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 60.8% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-y\right) \cdot z\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+208}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y) z)))
   (if (<= z -4.6e+139) t_0 (if (<= z 2.05e+208) (+ 1.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -y * z;
	double tmp;
	if (z <= -4.6e+139) {
		tmp = t_0;
	} else if (z <= 2.05e+208) {
		tmp = 1.0 + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -y * z
    if (z <= (-4.6d+139)) then
        tmp = t_0
    else if (z <= 2.05d+208) then
        tmp = 1.0d0 + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -y * z;
	double tmp;
	if (z <= -4.6e+139) {
		tmp = t_0;
	} else if (z <= 2.05e+208) {
		tmp = 1.0 + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -y * z
	tmp = 0
	if z <= -4.6e+139:
		tmp = t_0
	elif z <= 2.05e+208:
		tmp = 1.0 + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-y) * z)
	tmp = 0.0
	if (z <= -4.6e+139)
		tmp = t_0;
	elseif (z <= 2.05e+208)
		tmp = Float64(1.0 + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -y * z;
	tmp = 0.0;
	if (z <= -4.6e+139)
		tmp = t_0;
	elseif (z <= 2.05e+208)
		tmp = 1.0 + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-y) * z), $MachinePrecision]}, If[LessEqual[z, -4.6e+139], t$95$0, If[LessEqual[z, 2.05e+208], N[(1.0 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6e139 or 2.0499999999999999e208 < 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

      \[\leadsto \color{blue}{1 + \left(x + -1 \cdot \left(y \cdot z\right)\right)} \]
   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. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 34.8%

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

   if -4.6e139 < z < 2.0499999999999999e208

   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 68.4

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

   5. Applied rewrites 68.4%

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

Alternative 14: 61.1% 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 55.7

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

5. Applied rewrites 55.7%

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

Alternative 15: 21.2% 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 55.7

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

5. Applied rewrites 55.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 19.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(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))))