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

Percentage Accurate: 99.9% → 99.9%

Time: 11.9s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := x - t\_0\\ \mathbf{if}\;x \leq -0.98:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.84:\\ \;\;\;\;\cos y - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- x t_0)))
   (if (<= x -0.98) t_1 (if (<= x 0.84) (- (cos y) t_0) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = x - t_0;
	double tmp;
	if (x <= -0.98) {
		tmp = t_1;
	} else if (x <= 0.84) {
		tmp = cos(y) - t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = x - t_0
    if (x <= (-0.98d0)) then
        tmp = t_1
    else if (x <= 0.84d0) then
        tmp = cos(y) - t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double t_1 = x - t_0;
	double tmp;
	if (x <= -0.98) {
		tmp = t_1;
	} else if (x <= 0.84) {
		tmp = Math.cos(y) - t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = x - t_0
	tmp = 0
	if x <= -0.98:
		tmp = t_1
	elif x <= 0.84:
		tmp = math.cos(y) - t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(x - t_0)
	tmp = 0.0
	if (x <= -0.98)
		tmp = t_1;
	elseif (x <= 0.84)
		tmp = Float64(cos(y) - t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = x - t_0;
	tmp = 0.0;
	if (x <= -0.98)
		tmp = t_1;
	elseif (x <= 0.84)
		tmp = cos(y) - t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.98], t$95$1, If[LessEqual[x, 0.84], N[(N[Cos[y], $MachinePrecision] - t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.97999999999999998 or 0.839999999999999969 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 98.8%

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

   if -0.97999999999999998 < x < 0.839999999999999969

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\cos y}, \mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 99.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{*.f64}\left(\color{blue}{z}, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   5. Simplified 99.0%

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

Alternative 3: 93.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - z \cdot \sin y\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+36}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* z (sin y)))))
   (if (<= z -4.5e+25) t_0 (if (<= z 2.05e+36) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x - (z * sin(y));
	double tmp;
	if (z <= -4.5e+25) {
		tmp = t_0;
	} else if (z <= 2.05e+36) {
		tmp = x + cos(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 = x - (z * sin(y))
    if (z <= (-4.5d+25)) then
        tmp = t_0
    else if (z <= 2.05d+36) then
        tmp = x + cos(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 = x - (z * Math.sin(y));
	double tmp;
	if (z <= -4.5e+25) {
		tmp = t_0;
	} else if (z <= 2.05e+36) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (z * math.sin(y))
	tmp = 0
	if z <= -4.5e+25:
		tmp = t_0
	elif z <= 2.05e+36:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(z * sin(y)))
	tmp = 0.0
	if (z <= -4.5e+25)
		tmp = t_0;
	elseif (z <= 2.05e+36)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (z * sin(y));
	tmp = 0.0;
	if (z <= -4.5e+25)
		tmp = t_0;
	elseif (z <= 2.05e+36)
		tmp = x + cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.5e+25], t$95$0, If[LessEqual[z, 2.05e+36], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5000000000000003e25 or 2.05000000000000006e36 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 90.8%

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

   if -4.5000000000000003e25 < z < 2.05000000000000006e36

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\cos y, \color{blue}{x}\right) \]

      3. cos-lowering-cos.f64 98.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{cos.f64}\left(y\right), x\right) \]

   5. Simplified 98.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+25}:\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+36}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(0 - \sin y\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+52}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- 0.0 (sin y)))))
   (if (<= z -1.3e+132) t_0 (if (<= z 1.5e+52) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (0.0 - sin(y));
	double tmp;
	if (z <= -1.3e+132) {
		tmp = t_0;
	} else if (z <= 1.5e+52) {
		tmp = x + cos(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 * (0.0d0 - sin(y))
    if (z <= (-1.3d+132)) then
        tmp = t_0
    else if (z <= 1.5d+52) then
        tmp = x + cos(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 * (0.0 - Math.sin(y));
	double tmp;
	if (z <= -1.3e+132) {
		tmp = t_0;
	} else if (z <= 1.5e+52) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (0.0 - math.sin(y))
	tmp = 0
	if z <= -1.3e+132:
		tmp = t_0
	elif z <= 1.5e+52:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(0.0 - sin(y)))
	tmp = 0.0
	if (z <= -1.3e+132)
		tmp = t_0;
	elseif (z <= 1.5e+52)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (0.0 - sin(y));
	tmp = 0.0;
	if (z <= -1.3e+132)
		tmp = t_0;
	elseif (z <= 1.5e+52)
		tmp = x + cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(0.0 - N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.3e+132], t$95$0, If[LessEqual[z, 1.5e+52], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e132 or 1.5e52 < z

   1. Initial program 99.8%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(x + \cos y\right) + z \cdot \sin y}{\left(x + \cos y\right) \cdot \left(x + \cos y\right) - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\left(x + \cos y\right) \cdot \left(x + \cos y\right) - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}{\left(x + \cos y\right) + z \cdot \sin y}}}\right)\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\left(x + \cos y\right) - \color{blue}{z \cdot \sin y}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(x + \cos y\right) - z \cdot \sin y\right)}\right)\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(x + \color{blue}{\left(\cos y - z \cdot \sin y\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{\left(\cos y - z \cdot \sin y\right)}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(\cos y + \color{blue}{\left(\mathsf{neg}\left(z \cdot \sin y\right)\right)}\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(z \cdot \sin y\right)\right) + \color{blue}{\cos y}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(z \cdot \left(\mathsf{neg}\left(\sin y\right)\right) + \cos \color{blue}{y}\right)\right)\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{fma.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right)}, \cos y\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{x + \mathsf{fma}\left(z, 0 - \sin y, \cos y\right)}}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{z \cdot \sin y}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(z \cdot \sin y\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(z, \color{blue}{\sin y}\right)\right)\right) \]

      3. sin-lowering-sin.f64 66.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right)\right) \]

   7. Simplified 66.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{z \cdot \sin y}}} \]
   8. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. metadata-eval N/A

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

      3. neg-mul-1 N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\sin y\right)\right), \color{blue}{z}\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\sin y\right), z\right) \]

      8. sin-lowering-sin.f64 66.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(y\right)\right), z\right) \]

   9. Applied egg-rr 66.1%

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

   if -1.3e132 < z < 1.5e52

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\cos y, \color{blue}{x}\right) \]

      3. cos-lowering-cos.f64 91.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{cos.f64}\left(y\right), x\right) \]

   5. Simplified 91.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+132}:\\ \;\;\;\;z \cdot \left(0 - \sin y\right)\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+52}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(0 - \sin y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ \mathbf{if}\;y \leq -80000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, -0.5\right), 0\right) - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))))
   (if (<= y -80000.0)
     t_0
     (if (<= y 0.75)
       (+
        1.0
        (fma y (- (fma y (fma z (* y 0.16666666666666666) -0.5) 0.0) z) x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double tmp;
	if (y <= -80000.0) {
		tmp = t_0;
	} else if (y <= 0.75) {
		tmp = 1.0 + fma(y, (fma(y, fma(z, (y * 0.16666666666666666), -0.5), 0.0) - z), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	tmp = 0.0
	if (y <= -80000.0)
		tmp = t_0;
	elseif (y <= 0.75)
		tmp = Float64(1.0 + fma(y, Float64(fma(y, fma(z, Float64(y * 0.16666666666666666), -0.5), 0.0) - z), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -80000.0], t$95$0, If[LessEqual[y, 0.75], N[(1.0 + N[(y * N[(N[(y * N[(z * N[(y * 0.16666666666666666), $MachinePrecision] + -0.5), $MachinePrecision] + 0.0), $MachinePrecision] - z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8e4 or 0.75 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\cos y, \color{blue}{x}\right) \]

      3. cos-lowering-cos.f64 62.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{cos.f64}\left(y\right), x\right) \]

   5. Simplified 62.1%

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

   if -8e4 < y < 0.75

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + \color{blue}{x}\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)}, x\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right)\right), \color{blue}{z}\right), x\right)\right) \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) + 0\right), z\right), x\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right), 0\right), z\right), x\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 0\right), z\right), x\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot z + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 0\right), z\right), x\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \left(z \cdot \left(\frac{1}{6} \cdot y\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 0\right), z\right), x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \left(z \cdot \left(\frac{1}{6} \cdot y\right) + \frac{-1}{2}\right), 0\right), z\right), x\right)\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \left(\frac{1}{6} \cdot y\right), \frac{-1}{2}\right), 0\right), z\right), x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \left(y \cdot \frac{1}{6}\right), \frac{-1}{2}\right), 0\right), z\right), x\right)\right) \]

      13. *-lowering-*.f64 99.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \frac{1}{6}\right), \frac{-1}{2}\right), 0\right), z\right), x\right)\right) \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, -0.5\right), 0\right) - z, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -80000:\\ \;\;\;\;x + \cos y\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, -0.5\right), 0\right) - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-11}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-22}:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.5e-11) (+ x 1.0) (if (<= x 5.4e-22) (cos y) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e-11) {
		tmp = x + 1.0;
	} else if (x <= 5.4e-22) {
		tmp = cos(y);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-9.5d-11)) then
        tmp = x + 1.0d0
    else if (x <= 5.4d-22) then
        tmp = cos(y)
    else
        tmp = x + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e-11) {
		tmp = x + 1.0;
	} else if (x <= 5.4e-22) {
		tmp = Math.cos(y);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.5e-11:
		tmp = x + 1.0
	elif x <= 5.4e-22:
		tmp = math.cos(y)
	else:
		tmp = x + 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.5e-11)
		tmp = Float64(x + 1.0);
	elseif (x <= 5.4e-22)
		tmp = cos(y);
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.5e-11)
		tmp = x + 1.0;
	elseif (x <= 5.4e-22)
		tmp = cos(y);
	else
		tmp = x + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.5e-11], N[(x + 1.0), $MachinePrecision], If[LessEqual[x, 5.4e-22], N[Cos[y], $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.49999999999999951e-11 or 5.4000000000000004e-22 < 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. +-commutative N/A

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

      2. +-lowering-+.f64 81.9%

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{1}\right) \]

   5. Simplified 81.9%

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

   if -9.49999999999999951e-11 < x < 5.4000000000000004e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\cos y, \color{blue}{x}\right) \]

      3. cos-lowering-cos.f64 60.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{cos.f64}\left(y\right), x\right) \]

   5. Simplified 60.7%

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

   6. Taylor expanded in x around 0

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

      1. cos-lowering-cos.f64 60.7%

         \[\leadsto \mathsf{cos.f64}\left(y\right) \]

   8. Simplified 60.7%

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

Alternative 7: 70.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -85000000:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 98000:\\ \;\;\;\;1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, -0.5\right), 0\right) - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -85000000.0)
   (+ x 1.0)
   (if (<= y 98000.0)
     (+ 1.0 (fma y (- (fma y (fma z (* y 0.16666666666666666) -0.5) 0.0) z) x))
     (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -85000000.0) {
		tmp = x + 1.0;
	} else if (y <= 98000.0) {
		tmp = 1.0 + fma(y, (fma(y, fma(z, (y * 0.16666666666666666), -0.5), 0.0) - z), x);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -85000000.0)
		tmp = Float64(x + 1.0);
	elseif (y <= 98000.0)
		tmp = Float64(1.0 + fma(y, Float64(fma(y, fma(z, Float64(y * 0.16666666666666666), -0.5), 0.0) - z), x));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -85000000.0], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 98000.0], N[(1.0 + N[(y * N[(N[(y * N[(z * N[(y * 0.16666666666666666), $MachinePrecision] + -0.5), $MachinePrecision] + 0.0), $MachinePrecision] - z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5e7 or 98000 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 40.0%

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{1}\right) \]

   5. Simplified 40.0%

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

   if -8.5e7 < y < 98000

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + \color{blue}{x}\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)}, x\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right)\right), \color{blue}{z}\right), x\right)\right) \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) + 0\right), z\right), x\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right), 0\right), z\right), x\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 0\right), z\right), x\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot z + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 0\right), z\right), x\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \left(z \cdot \left(\frac{1}{6} \cdot y\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 0\right), z\right), x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \left(z \cdot \left(\frac{1}{6} \cdot y\right) + \frac{-1}{2}\right), 0\right), z\right), x\right)\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \left(\frac{1}{6} \cdot y\right), \frac{-1}{2}\right), 0\right), z\right), x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \left(y \cdot \frac{1}{6}\right), \frac{-1}{2}\right), 0\right), z\right), x\right)\right) \]

      13. *-lowering-*.f64 99.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{*.f64}\left(y, \frac{1}{6}\right), \frac{-1}{2}\right), 0\right), z\right), x\right)\right) \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(z, y \cdot 0.16666666666666666, -0.5\right), 0\right) - z, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -60000:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 30000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.5, 0\right) - z, x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -60000.0)
   (+ x 1.0)
   (if (<= y 30000.0) (fma y (- (fma y -0.5 0.0) z) (+ x 1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -60000.0) {
		tmp = x + 1.0;
	} else if (y <= 30000.0) {
		tmp = fma(y, (fma(y, -0.5, 0.0) - z), (x + 1.0));
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -60000.0)
		tmp = Float64(x + 1.0);
	elseif (y <= 30000.0)
		tmp = fma(y, Float64(fma(y, -0.5, 0.0) - z), Float64(x + 1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -60000.0], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 30000.0], N[(y * N[(N[(y * -0.5 + 0.0), $MachinePrecision] - z), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6e4 or 3e4 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 39.7%

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{1}\right) \]

   5. Simplified 39.7%

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

   if -6e4 < y < 3e4

   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

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{\left(\frac{-1}{2} \cdot y - z\right)}, \left(1 + x\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{-1}{2} \cdot y\right), \color{blue}{z}\right), \left(1 + x\right)\right) \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{-1}{2} \cdot y + 0\right), z\right), \left(1 + x\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\left(y \cdot \frac{-1}{2} + 0\right), z\right), \left(1 + x\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \frac{-1}{2}, 0\right), z\right), \left(1 + x\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \frac{-1}{2}, 0\right), z\right), \left(x + 1\right)\right) \]

      9. +-lowering-+.f64 99.8%

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(y, \frac{-1}{2}, 0\right), z\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.5, 0\right) - z, x + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+34}:\\ \;\;\;\;x + 1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+19}:\\ \;\;\;\;x - \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.65e+34)
   (+ x 1.0)
   (if (<= y 1.7e+19) (- x (fma y z -1.0)) (+ x 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.65e+34) {
		tmp = x + 1.0;
	} else if (y <= 1.7e+19) {
		tmp = x - fma(y, z, -1.0);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.65e+34)
		tmp = Float64(x + 1.0);
	elseif (y <= 1.7e+19)
		tmp = Float64(x - fma(y, z, -1.0));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.65e+34], N[(x + 1.0), $MachinePrecision], If[LessEqual[y, 1.7e+19], N[(x - N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.64999999999999994e34 or 1.7e19 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.2%

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{1}\right) \]

   5. Simplified 42.2%

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

   if -1.64999999999999994e34 < y < 1.7e19

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

         \[\leadsto \left(x - y \cdot z\right) + 1 \]

      4. associate-+l- N/A

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

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(y \cdot z - 1\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(y \cdot z + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(y \cdot z + -1\right)\right) \]

      8. accelerator-lowering-fma.f64 91.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{fma.f64}\left(y, \color{blue}{z}, -1\right)\right) \]

   5. Simplified 91.5%

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

Alternative 10: 64.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - y \cdot z\\ \mathbf{if}\;z \leq -6.1 \cdot 10^{+206}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+276}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* y z))))
   (if (<= z -6.1e+206) t_0 (if (<= z 1.95e+276) (+ x 1.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x - (y * z);
	double tmp;
	if (z <= -6.1e+206) {
		tmp = t_0;
	} else if (z <= 1.95e+276) {
		tmp = x + 1.0;
	} 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 = x - (y * z)
    if (z <= (-6.1d+206)) then
        tmp = t_0
    else if (z <= 1.95d+276) then
        tmp = x + 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (y * z);
	double tmp;
	if (z <= -6.1e+206) {
		tmp = t_0;
	} else if (z <= 1.95e+276) {
		tmp = x + 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (y * z)
	tmp = 0
	if z <= -6.1e+206:
		tmp = t_0
	elif z <= 1.95e+276:
		tmp = x + 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(y * z))
	tmp = 0.0
	if (z <= -6.1e+206)
		tmp = t_0;
	elseif (z <= 1.95e+276)
		tmp = Float64(x + 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (y * z);
	tmp = 0.0;
	if (z <= -6.1e+206)
		tmp = t_0;
	elseif (z <= 1.95e+276)
		tmp = x + 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.1e+206], t$95$0, If[LessEqual[z, 1.95e+276], N[(x + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.09999999999999967e206 or 1.9500000000000001e276 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 98.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{y}\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 63.9%

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

   if -6.09999999999999967e206 < z < 1.9500000000000001e276

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 66.0%

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{1}\right) \]

   5. Simplified 66.0%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{+206}:\\ \;\;\;\;x - y \cdot z\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+276}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.5% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+208}:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.2e+208) (- 0.0 (* y z)) (+ x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.2e+208) {
		tmp = 0.0 - (y * z);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-7.2d+208)) then
        tmp = 0.0d0 - (y * z)
    else
        tmp = x + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.2e+208) {
		tmp = 0.0 - (y * z);
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.2e+208:
		tmp = 0.0 - (y * z)
	else:
		tmp = x + 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.2e+208)
		tmp = Float64(0.0 - Float64(y * z));
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.2e+208)
		tmp = 0.0 - (y * z);
	else
		tmp = x + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.2e+208], N[(0.0 - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.20000000000000005e208

   1. Initial program 100.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(x + \cos y\right) + z \cdot \sin y}{\left(x + \cos y\right) \cdot \left(x + \cos y\right) - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\left(x + \cos y\right) \cdot \left(x + \cos y\right) - \left(z \cdot \sin y\right) \cdot \left(z \cdot \sin y\right)}{\left(x + \cos y\right) + z \cdot \sin y}}}\right)\right) \]

      5. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\left(x + \cos y\right) - \color{blue}{z \cdot \sin y}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\left(x + \cos y\right) - z \cdot \sin y\right)}\right)\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(x + \color{blue}{\left(\cos y - z \cdot \sin y\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{\left(\cos y - z \cdot \sin y\right)}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(\cos y + \color{blue}{\left(\mathsf{neg}\left(z \cdot \sin y\right)\right)}\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(\left(\mathsf{neg}\left(z \cdot \sin y\right)\right) + \color{blue}{\cos y}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \left(z \cdot \left(\mathsf{neg}\left(\sin y\right)\right) + \cos \color{blue}{y}\right)\right)\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \mathsf{fma.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\sin y\right)\right)}, \cos y\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{x + \mathsf{fma}\left(z, 0 - \sin y, \cos y\right)}}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{z \cdot \sin y}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left(z \cdot \sin y\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(z, \color{blue}{\sin y}\right)\right)\right) \]

      3. sin-lowering-sin.f64 82.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(z, \mathsf{sin.f64}\left(y\right)\right)\right)\right) \]

   7. Simplified 82.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{z \cdot \sin y}}} \]
   8. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. metadata-eval N/A

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

      3. neg-mul-1 N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\sin y\right)\right), \color{blue}{z}\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\sin y\right), z\right) \]

      8. sin-lowering-sin.f64 83.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(y\right)\right), z\right) \]

   9. Applied egg-rr 83.0%

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

   10. Taylor expanded in y around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\color{blue}{y}\right), z\right) \]
   11. Step-by-step derivation

   12. Simplified 48.1%

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

   if -7.20000000000000005e208 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 64.3%

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{1}\right) \]

   5. Simplified 64.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+208}:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.8% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x -7e-17) x (if (<= x 1.0) 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7e-17) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-7d-17)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7e-17) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7e-17:
		tmp = x
	elif x <= 1.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7e-17)
		tmp = x;
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7e-17)
		tmp = x;
	elseif (x <= 1.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7e-17], x, If[LessEqual[x, 1.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.0000000000000003e-17 or 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 80.6%

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

   if -7.0000000000000003e-17 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 40.2%

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{1}\right) \]

   5. Simplified 40.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 39.4%

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

Alternative 13: 61.8% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 60.9%

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{1}\right) \]

5. Simplified 60.9%

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

Alternative 14: 21.5% accurate, 212.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 60.9%

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{1}\right) \]

5. Simplified 60.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 20.9%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024193 
(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))))