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

Percentage Accurate: 99.9% → 99.9%

Time: 9.5s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 81.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y + \left(x + \sin y\right)\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;x + z\\ \mathbf{elif}\;t\_0 \leq -0.002:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 0.0005:\\ \;\;\;\;y + \left(x + z\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z (cos y)) (+ x (sin y)))))
   (if (<= t_0 -50.0)
     (+ x z)
     (if (<= t_0 -0.002)
       (sin y)
       (if (<= t_0 0.0005) (+ y (+ x z)) (if (<= t_0 1.0) (sin y) (+ x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * cos(y)) + (x + sin(y));
	double tmp;
	if (t_0 <= -50.0) {
		tmp = x + z;
	} else if (t_0 <= -0.002) {
		tmp = sin(y);
	} else if (t_0 <= 0.0005) {
		tmp = y + (x + z);
	} else if (t_0 <= 1.0) {
		tmp = sin(y);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * cos(y)) + (x + sin(y))
    if (t_0 <= (-50.0d0)) then
        tmp = x + z
    else if (t_0 <= (-0.002d0)) then
        tmp = sin(y)
    else if (t_0 <= 0.0005d0) then
        tmp = y + (x + z)
    else if (t_0 <= 1.0d0) then
        tmp = sin(y)
    else
        tmp = x + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * Math.cos(y)) + (x + Math.sin(y));
	double tmp;
	if (t_0 <= -50.0) {
		tmp = x + z;
	} else if (t_0 <= -0.002) {
		tmp = Math.sin(y);
	} else if (t_0 <= 0.0005) {
		tmp = y + (x + z);
	} else if (t_0 <= 1.0) {
		tmp = Math.sin(y);
	} else {
		tmp = x + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * math.cos(y)) + (x + math.sin(y))
	tmp = 0
	if t_0 <= -50.0:
		tmp = x + z
	elif t_0 <= -0.002:
		tmp = math.sin(y)
	elif t_0 <= 0.0005:
		tmp = y + (x + z)
	elif t_0 <= 1.0:
		tmp = math.sin(y)
	else:
		tmp = x + z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * cos(y)) + Float64(x + sin(y)))
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = Float64(x + z);
	elseif (t_0 <= -0.002)
		tmp = sin(y);
	elseif (t_0 <= 0.0005)
		tmp = Float64(y + Float64(x + z));
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = Float64(x + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * cos(y)) + (x + sin(y));
	tmp = 0.0;
	if (t_0 <= -50.0)
		tmp = x + z;
	elseif (t_0 <= -0.002)
		tmp = sin(y);
	elseif (t_0 <= 0.0005)
		tmp = y + (x + z);
	elseif (t_0 <= 1.0)
		tmp = sin(y);
	else
		tmp = x + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], N[(x + z), $MachinePrecision], If[LessEqual[t$95$0, -0.002], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 0.0005], N[(y + N[(x + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[y], $MachinePrecision], N[(x + z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -50 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if -50 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -2e-3 or 5.0000000000000001e-4 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 93.9

         \[\leadsto \color{blue}{\sin y} \]

   8. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\sin y} \]

   if -2e-3 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \cos y + \left(x + \sin y\right) \leq -50:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \cdot \cos y + \left(x + \sin y\right) \leq -0.002:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;z \cdot \cos y + \left(x + \sin y\right) \leq 0.0005:\\ \;\;\;\;y + \left(x + z\right)\\ \mathbf{elif}\;z \cdot \cos y + \left(x + \sin y\right) \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, z \cdot \frac{\cos y}{x}, x\right)\\ \mathbf{if}\;x \leq -2.25 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-82}:\\ \;\;\;\;\sin y + z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x (* z (/ (cos y) x)) x)))
   (if (<= x -2.25e-31)
     t_0
     (if (<= x 9.5e-82) (+ (sin y) (* z (cos y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, (z * (cos(y) / x)), x);
	double tmp;
	if (x <= -2.25e-31) {
		tmp = t_0;
	} else if (x <= 9.5e-82) {
		tmp = sin(y) + (z * cos(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, Float64(z * Float64(cos(y) / x)), x)
	tmp = 0.0
	if (x <= -2.25e-31)
		tmp = t_0;
	elseif (x <= 9.5e-82)
		tmp = Float64(sin(y) + Float64(z * cos(y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z * N[(N[Cos[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -2.25e-31], t$95$0, If[LessEqual[x, 9.5e-82], N[(N[Sin[y], $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2500000000000002e-31 or 9.4999999999999996e-82 < x

   1. Initial program 99.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. neg-mul-1 N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

          \[\leadsto x \cdot \color{blue}{\frac{\sin y + z \cdot \cos y}{x}} + x \]

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 96.7

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

   10. Applied rewrites 96.7%

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

   if -2.2500000000000002e-31 < x < 9.4999999999999996e-82

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-sin.f64 95.3

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

   5. Applied rewrites 95.3%

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

Alternative 4: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, z \cdot \frac{\cos y}{x}, x\right)\\ \mathbf{if}\;x \leq -2.25 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(z, \cos y, \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x (* z (/ (cos y) x)) x)))
   (if (<= x -2.25e-31) t_0 (if (<= x 9.5e-82) (fma z (cos y) (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, (z * (cos(y) / x)), x);
	double tmp;
	if (x <= -2.25e-31) {
		tmp = t_0;
	} else if (x <= 9.5e-82) {
		tmp = fma(z, cos(y), sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, Float64(z * Float64(cos(y) / x)), x)
	tmp = 0.0
	if (x <= -2.25e-31)
		tmp = t_0;
	elseif (x <= 9.5e-82)
		tmp = fma(z, cos(y), sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z * N[(N[Cos[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -2.25e-31], t$95$0, If[LessEqual[x, 9.5e-82], N[(z * N[Cos[y], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2500000000000002e-31 or 9.4999999999999996e-82 < x

   1. Initial program 99.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. neg-mul-1 N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

          \[\leadsto x \cdot \color{blue}{\frac{\sin y + z \cdot \cos y}{x}} + x \]

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 96.7

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

   10. Applied rewrites 96.7%

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

   if -2.2500000000000002e-31 < x < 9.4999999999999996e-82

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 95.3

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

   5. Applied rewrites 95.3%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 6: 89.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := \mathsf{fma}\left(x, z \cdot \frac{\cos y}{x}, x\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-40}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))) (t_1 (fma x (* z (/ (cos y) x)) x)))
   (if (<= z -1.3e+148)
     t_0
     (if (<= z -5.2e-19)
       t_1
       (if (<= z 2.1e-40) (+ x (sin y)) (if (<= z 5.6e+131) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double t_1 = fma(x, (z * (cos(y) / x)), x);
	double tmp;
	if (z <= -1.3e+148) {
		tmp = t_0;
	} else if (z <= -5.2e-19) {
		tmp = t_1;
	} else if (z <= 2.1e-40) {
		tmp = x + sin(y);
	} else if (z <= 5.6e+131) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	t_1 = fma(x, Float64(z * Float64(cos(y) / x)), x)
	tmp = 0.0
	if (z <= -1.3e+148)
		tmp = t_0;
	elseif (z <= -5.2e-19)
		tmp = t_1;
	elseif (z <= 2.1e-40)
		tmp = Float64(x + sin(y));
	elseif (z <= 5.6e+131)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(z * N[(N[Cos[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.3e+148], t$95$0, If[LessEqual[z, -5.2e-19], t$95$1, If[LessEqual[z, 2.1e-40], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.6e+131], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e148 or 5.6000000000000001e131 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -1.3e148 < z < -5.20000000000000026e-19 or 2.10000000000000018e-40 < z < 5.6000000000000001e131

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. neg-mul-1 N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

          \[\leadsto x \cdot \color{blue}{\frac{\sin y + z \cdot \cos y}{x}} + x \]

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 92.5%

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

   8. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 89.2

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

   10. Applied rewrites 89.2%

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

   if -5.20000000000000026e-19 < z < 2.10000000000000018e-40

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 94.8

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

   5. Applied rewrites 94.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+148}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(x, z \cdot \frac{\cos y}{x}, x\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-40}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(x, z \cdot \frac{\cos y}{x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+108}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -6.5e+33)
     t_0
     (if (<= z 2.2e-40) (+ x (sin y)) (if (<= z 4.4e+108) (+ x z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -6.5e+33) {
		tmp = t_0;
	} else if (z <= 2.2e-40) {
		tmp = x + sin(y);
	} else if (z <= 4.4e+108) {
		tmp = x + z;
	} 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 = z * cos(y)
    if (z <= (-6.5d+33)) then
        tmp = t_0
    else if (z <= 2.2d-40) then
        tmp = x + sin(y)
    else if (z <= 4.4d+108) then
        tmp = x + z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.cos(y);
	double tmp;
	if (z <= -6.5e+33) {
		tmp = t_0;
	} else if (z <= 2.2e-40) {
		tmp = x + Math.sin(y);
	} else if (z <= 4.4e+108) {
		tmp = x + z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -6.5e+33:
		tmp = t_0
	elif z <= 2.2e-40:
		tmp = x + math.sin(y)
	elif z <= 4.4e+108:
		tmp = x + z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -6.5e+33)
		tmp = t_0;
	elseif (z <= 2.2e-40)
		tmp = Float64(x + sin(y));
	elseif (z <= 4.4e+108)
		tmp = Float64(x + z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * cos(y);
	tmp = 0.0;
	if (z <= -6.5e+33)
		tmp = t_0;
	elseif (z <= 2.2e-40)
		tmp = x + sin(y);
	elseif (z <= 4.4e+108)
		tmp = x + z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5e+33], t$95$0, If[LessEqual[z, 2.2e-40], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e+108], N[(x + z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.49999999999999993e33 or 4.4000000000000003e108 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if -6.49999999999999993e33 < z < 2.20000000000000009e-40

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if 2.20000000000000009e-40 < z < 4.4000000000000003e108

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+33}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-40}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+108}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sin y\\ \mathbf{if}\;y \leq -0.195:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.235:\\ \;\;\;\;z + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot -0.5, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (sin y))))
   (if (<= y -0.195)
     t_0
     (if (<= y 0.235) (+ z (fma y (fma y (* z -0.5) 1.0) x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + sin(y);
	double tmp;
	if (y <= -0.195) {
		tmp = t_0;
	} else if (y <= 0.235) {
		tmp = z + fma(y, fma(y, (z * -0.5), 1.0), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + sin(y))
	tmp = 0.0
	if (y <= -0.195)
		tmp = t_0;
	elseif (y <= 0.235)
		tmp = Float64(z + fma(y, fma(y, Float64(z * -0.5), 1.0), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.195], t$95$0, If[LessEqual[y, 0.235], N[(z + N[(y * N[(y * N[(z * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.19500000000000001 or 0.23499999999999999 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sin.f64 52.7

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

   5. Applied rewrites 52.7%

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

   if -0.19500000000000001 < y < 0.23499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.195:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;y \leq 0.235:\\ \;\;\;\;z + \mathsf{fma}\left(y, \mathsf{fma}\left(y, z \cdot -0.5, 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.9% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 64.7

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

5. Applied rewrites 64.7%

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

6. Final simplification 64.7%

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

Alternative 10: 37.4% accurate, 53.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + 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 + y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 54.4

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

5. Applied rewrites 54.4%

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

6. Taylor expanded in z around 0

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

   1. lower-+.f64 34.4

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

8. Applied rewrites 34.4%

   \[\leadsto \color{blue}{x + y} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C"
  :precision binary64
  (+ (+ x (sin y)) (* z (cos y))))