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

Percentage Accurate: 99.9% → 99.9%

Time: 7.7s

Alternatives: 9

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

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

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-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 81.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y + \left(x + \sin y\right)\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z (cos y)) (+ x (sin y)))))
   (if (<= t_0 -1000.0)
     (+ x z)
     (if (<= t_0 -0.1)
       (sin y)
       (if (<= t_0 5e-9) (+ (+ x y) 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 <= -1000.0) {
		tmp = x + z;
	} else if (t_0 <= -0.1) {
		tmp = sin(y);
	} else if (t_0 <= 5e-9) {
		tmp = (x + y) + 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 <= (-1000.0d0)) then
        tmp = x + z
    else if (t_0 <= (-0.1d0)) then
        tmp = sin(y)
    else if (t_0 <= 5d-9) then
        tmp = (x + y) + 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 <= -1000.0) {
		tmp = x + z;
	} else if (t_0 <= -0.1) {
		tmp = Math.sin(y);
	} else if (t_0 <= 5e-9) {
		tmp = (x + y) + 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 <= -1000.0:
		tmp = x + z
	elif t_0 <= -0.1:
		tmp = math.sin(y)
	elif t_0 <= 5e-9:
		tmp = (x + y) + 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 <= -1000.0)
		tmp = Float64(x + z);
	elseif (t_0 <= -0.1)
		tmp = sin(y);
	elseif (t_0 <= 5e-9)
		tmp = Float64(Float64(x + y) + 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 <= -1000.0)
		tmp = x + z;
	elseif (t_0 <= -0.1)
		tmp = sin(y);
	elseif (t_0 <= 5e-9)
		tmp = (x + y) + 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, -1000.0], N[(x + z), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[y], $MachinePrecision], If[LessEqual[t$95$0, 5e-9], N[(N[(x + y), $MachinePrecision] + z), $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))) < -1e3 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   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 73.6

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

   5. Applied rewrites 73.6%

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

   if -1e3 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.10000000000000001 or 5.0000000000000001e-9 < (+.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 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \sin y \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \sin y \]

   if -0.10000000000000001 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 5.0000000000000001e-9

   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. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \cos y + \left(x + \sin y\right) \leq -1000:\\ \;\;\;\;x + z\\ \mathbf{elif}\;z \cdot \cos y + \left(x + \sin y\right) \leq -0.1:\\ \;\;\;\;\sin y\\ \mathbf{elif}\;z \cdot \cos y + \left(x + \sin y\right) \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(x + y\right) + z\\ \mathbf{elif}\;z \cdot \cos y + \left(x + \sin y\right) \leq 1:\\ \;\;\;\;\sin y\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{z \cdot \cos y}{x}, x, x\right)\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ (* z (cos y)) x) x x)))
   (if (<= x -5.8e-21) t_0 (if (<= x 3.7e-20) (fma (cos y) z (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(((z * cos(y)) / x), x, x);
	double tmp;
	if (x <= -5.8e-21) {
		tmp = t_0;
	} else if (x <= 3.7e-20) {
		tmp = fma(cos(y), z, sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(Float64(z * cos(y)) / x), x, x)
	tmp = 0.0
	if (x <= -5.8e-21)
		tmp = t_0;
	elseif (x <= 3.7e-20)
		tmp = fma(cos(y), z, sin(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[x, -5.8e-21], t$95$0, If[LessEqual[x, 3.7e-20], N[(N[Cos[y], $MachinePrecision] * z + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e-21 or 3.7000000000000001e-20 < 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-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. flip3-+ N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-pow.f64 N/A

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 99.0%

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

   if -5.8e-21 < x < 3.7000000000000001e-20

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 94.6

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

   5. Applied rewrites 94.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot \cos y}{x}, x, x\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, \sin y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot \cos y}{x}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := \mathsf{fma}\left(\frac{t\_0}{x}, x, x\right)\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-26}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))) (t_1 (fma (/ t_0 x) x x)))
   (if (<= z -9.2e+224)
     (fma (cos y) z (+ x y))
     (if (<= z -1.65e-31)
       t_1
       (if (<= z 1.3e-26) (+ x (sin y)) (if (<= z 9.5e+92) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double t_1 = fma((t_0 / x), x, x);
	double tmp;
	if (z <= -9.2e+224) {
		tmp = fma(cos(y), z, (x + y));
	} else if (z <= -1.65e-31) {
		tmp = t_1;
	} else if (z <= 1.3e-26) {
		tmp = x + sin(y);
	} else if (z <= 9.5e+92) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	t_1 = fma(Float64(t_0 / x), x, x)
	tmp = 0.0
	if (z <= -9.2e+224)
		tmp = fma(cos(y), z, Float64(x + y));
	elseif (z <= -1.65e-31)
		tmp = t_1;
	elseif (z <= 1.3e-26)
		tmp = Float64(x + sin(y));
	elseif (z <= 9.5e+92)
		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[(N[(t$95$0 / x), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[z, -9.2e+224], N[(N[Cos[y], $MachinePrecision] * z + N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.65e-31], t$95$1, If[LessEqual[z, 1.3e-26], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e+92], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.20000000000000079e224

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.8

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.2

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

   7. Applied rewrites 98.2%

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

   if -9.20000000000000079e224 < z < -1.65e-31 or 1.30000000000000005e-26 < z < 9.4999999999999995e92

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. flip3-+ N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-pow.f64 N/A

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 90.0

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

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 88.1%

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

   if -1.65e-31 < z < 1.30000000000000005e-26

   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 96.5

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

   5. Applied rewrites 96.5%

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

   if 9.4999999999999995e92 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 90.4

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

   5. Applied rewrites 90.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x + y\right)\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot \cos y}{x}, x, x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-26}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot \cos y}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+47}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (cos y))))
   (if (<= z -3.2e+43) t_0 (if (<= z 2.7e+47) (+ x (sin y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * cos(y);
	double tmp;
	if (z <= -3.2e+43) {
		tmp = t_0;
	} else if (z <= 2.7e+47) {
		tmp = x + sin(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 = z * cos(y)
    if (z <= (-3.2d+43)) then
        tmp = t_0
    else if (z <= 2.7d+47) then
        tmp = x + sin(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 = z * Math.cos(y);
	double tmp;
	if (z <= -3.2e+43) {
		tmp = t_0;
	} else if (z <= 2.7e+47) {
		tmp = x + Math.sin(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.cos(y)
	tmp = 0
	if z <= -3.2e+43:
		tmp = t_0
	elif z <= 2.7e+47:
		tmp = x + math.sin(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * cos(y))
	tmp = 0.0
	if (z <= -3.2e+43)
		tmp = t_0;
	elseif (z <= 2.7e+47)
		tmp = Float64(x + sin(y));
	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 <= -3.2e+43)
		tmp = t_0;
	elseif (z <= 2.7e+47)
		tmp = x + sin(y);
	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, -3.2e+43], t$95$0, If[LessEqual[z, 2.7e+47], N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.20000000000000014e43 or 2.69999999999999996e47 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -3.20000000000000014e43 < z < 2.69999999999999996e47

   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 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+43}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+47}:\\ \;\;\;\;x + \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sin y\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.043:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (sin y))))
   (if (<= y -1.7e+19)
     t_0
     (if (<= y 0.043) (fma (fma (* -0.5 y) z 1.0) y (+ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + sin(y);
	double tmp;
	if (y <= -1.7e+19) {
		tmp = t_0;
	} else if (y <= 0.043) {
		tmp = fma(fma((-0.5 * y), z, 1.0), y, (x + z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + sin(y))
	tmp = 0.0
	if (y <= -1.7e+19)
		tmp = t_0;
	elseif (y <= 0.043)
		tmp = fma(fma(Float64(-0.5 * y), z, 1.0), y, Float64(x + z));
	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, -1.7e+19], t$95$0, If[LessEqual[y, 0.043], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e19 or 0.042999999999999997 < 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 61.3

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

   5. Applied rewrites 61.3%

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

   if -1.7e19 < y < 0.042999999999999997

   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

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+19}:\\ \;\;\;\;x + \sin y\\ \mathbf{elif}\;y \leq 0.043:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+19}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+19)
   (+ x z)
   (if (<= y 7.5e+14) (fma (fma (* -0.5 y) z 1.0) y (+ x z)) (+ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+19) {
		tmp = x + z;
	} else if (y <= 7.5e+14) {
		tmp = fma(fma((-0.5 * y), z, 1.0), y, (x + z));
	} else {
		tmp = x + z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.8e+19)
		tmp = Float64(x + z);
	elseif (y <= 7.5e+14)
		tmp = fma(fma(Float64(-0.5 * y), z, 1.0), y, Float64(x + z));
	else
		tmp = Float64(x + z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.8e+19], N[(x + z), $MachinePrecision], If[LessEqual[y, 7.5e+14], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e19 or 7.5e14 < 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 40.7

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

   5. Applied rewrites 40.7%

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

   if -1.8e19 < y < 7.5e14

   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

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+19}:\\ \;\;\;\;x + z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, 1\right), y, x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% 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 63.8

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

5. Applied rewrites 63.8%

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

6. Final simplification 63.8%

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

Alternative 9: 38.9% 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 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 54.3

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

5. Applied rewrites 54.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 30.5%

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

9. Final simplification 30.5%

   \[\leadsto x + y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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))))