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

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 99.9

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9000000:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9000000.0)
   (fma (sin y) (- z) (+ 1.0 x))
   (if (<= x 7.2e-46)
     (fma (sin y) (- z) (cos y))
     (- (+ 1.0 x) (* z (sin y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9000000.0) {
		tmp = fma(sin(y), -z, (1.0 + x));
	} else if (x <= 7.2e-46) {
		tmp = fma(sin(y), -z, cos(y));
	} else {
		tmp = (1.0 + x) - (z * sin(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9000000.0)
		tmp = fma(sin(y), Float64(-z), Float64(1.0 + x));
	elseif (x <= 7.2e-46)
		tmp = fma(sin(y), Float64(-z), cos(y));
	else
		tmp = Float64(Float64(1.0 + x) - Float64(z * sin(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9000000.0], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-46], N[(N[Sin[y], $MachinePrecision] * (-z) + N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9e6

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.9%

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

   if -9e6 < x < 7.2e-46

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-cos.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if 7.2e-46 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9000000:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ \mathbf{if}\;x \leq -9000000:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\cos y - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (if (<= x -9000000.0)
     (fma (sin y) (- z) (+ 1.0 x))
     (if (<= x 7.2e-46) (- (cos y) t_0) (- (+ 1.0 x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double tmp;
	if (x <= -9000000.0) {
		tmp = fma(sin(y), -z, (1.0 + x));
	} else if (x <= 7.2e-46) {
		tmp = cos(y) - t_0;
	} else {
		tmp = (1.0 + x) - t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	tmp = 0.0
	if (x <= -9000000.0)
		tmp = fma(sin(y), Float64(-z), Float64(1.0 + x));
	elseif (x <= 7.2e-46)
		tmp = Float64(cos(y) - t_0);
	else
		tmp = Float64(Float64(1.0 + x) - t_0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9000000.0], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-46], N[(N[Cos[y], $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9e6

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.9%

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

   if -9e6 < x < 7.2e-46

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-cos.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 7.2e-46 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9000000:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 5: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \mathbf{if}\;z \leq -265:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.4:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (sin y) (- z) (+ 1.0 x))))
   (if (<= z -265.0) t_0 (if (<= z 1.4) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(sin(y), -z, (1.0 + x));
	double tmp;
	if (z <= -265.0) {
		tmp = t_0;
	} else if (z <= 1.4) {
		tmp = x + cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(sin(y), Float64(-z), Float64(1.0 + x))
	tmp = 0.0
	if (z <= -265.0)
		tmp = t_0;
	elseif (z <= 1.4)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -265.0], t$95$0, If[LessEqual[z, 1.4], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -265 or 1.3999999999999999 < z

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.2%

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

   if -265 < z < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -265:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \mathbf{elif}\;z \leq 1.4:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + x\right) - z \cdot \sin y\\ \mathbf{if}\;z \leq -265:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.4:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ 1.0 x) (* z (sin y)))))
   (if (<= z -265.0) t_0 (if (<= z 1.4) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 + x) - (z * sin(y));
	double tmp;
	if (z <= -265.0) {
		tmp = t_0;
	} else if (z <= 1.4) {
		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 = (1.0d0 + x) - (z * sin(y))
    if (z <= (-265.0d0)) then
        tmp = t_0
    else if (z <= 1.4d0) 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 = (1.0 + x) - (z * Math.sin(y));
	double tmp;
	if (z <= -265.0) {
		tmp = t_0;
	} else if (z <= 1.4) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 + x) - (z * math.sin(y))
	tmp = 0
	if z <= -265.0:
		tmp = t_0
	elif z <= 1.4:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 + x) - Float64(z * sin(y)))
	tmp = 0.0
	if (z <= -265.0)
		tmp = t_0;
	elseif (z <= 1.4)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 + x) - (z * sin(y));
	tmp = 0.0;
	if (z <= -265.0)
		tmp = t_0;
	elseif (z <= 1.4)
		tmp = x + cos(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -265.0], t$95$0, If[LessEqual[z, 1.4], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -265 or 1.3999999999999999 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.2%

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

   if -265 < z < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -265:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 1.4:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+183}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+83}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))))
   (if (<= z -1.9e+183) t_0 (if (<= z 2e+83) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double tmp;
	if (z <= -1.9e+183) {
		tmp = t_0;
	} else if (z <= 2e+83) {
		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 * sin(y)
    if (z <= (-1.9d+183)) then
        tmp = t_0
    else if (z <= 2d+83) 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 * Math.sin(y);
	double tmp;
	if (z <= -1.9e+183) {
		tmp = t_0;
	} else if (z <= 2e+83) {
		tmp = x + Math.cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * math.sin(y)
	tmp = 0
	if z <= -1.9e+183:
		tmp = t_0
	elif z <= 2e+83:
		tmp = x + math.cos(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * sin(y))
	tmp = 0.0
	if (z <= -1.9e+183)
		tmp = t_0;
	elseif (z <= 2e+83)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * sin(y);
	tmp = 0.0;
	if (z <= -1.9e+183)
		tmp = t_0;
	elseif (z <= 2e+83)
		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[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.9e+183], t$95$0, If[LessEqual[z, 2e+83], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9e183 or 2.00000000000000006e83 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 70.8

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

   5. Applied rewrites 70.8%

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

   if -1.9e183 < z < 2.00000000000000006e83

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+183}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+83}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ \mathbf{if}\;y \leq -0.0026:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1100000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot y\right) \cdot 0.16666666666666666\right) \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))))
   (if (<= y -0.0026)
     t_0
     (if (<= y 1100000000000.0)
       (fma (- (* (* (* z y) 0.16666666666666666) y) z) y (+ 1.0 x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double tmp;
	if (y <= -0.0026) {
		tmp = t_0;
	} else if (y <= 1100000000000.0) {
		tmp = fma(((((z * y) * 0.16666666666666666) * y) - z), y, (1.0 + x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	tmp = 0.0
	if (y <= -0.0026)
		tmp = t_0;
	elseif (y <= 1100000000000.0)
		tmp = fma(Float64(Float64(Float64(Float64(z * y) * 0.16666666666666666) * y) - z), y, Float64(1.0 + 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, -0.0026], t$95$0, If[LessEqual[y, 1100000000000.0], N[(N[(N[(N[(N[(z * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0025999999999999999 or 1.1e12 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if -0.0025999999999999999 < y < 1.1e12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right)\right) \cdot y - z, y, 1 + x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0026:\\ \;\;\;\;x + \cos y\\ \mathbf{elif}\;y \leq 1100000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot y\right) \cdot 0.16666666666666666\right) \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.1:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5\right) \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.1)
   (+ 1.0 x)
   (if (<= y 1.45e+18)
     (fma (- (* (fma 0.16666666666666666 (* z y) -0.5) y) z) y (+ 1.0 x))
     (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.1) {
		tmp = 1.0 + x;
	} else if (y <= 1.45e+18) {
		tmp = fma(((fma(0.16666666666666666, (z * y), -0.5) * y) - z), y, (1.0 + x));
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.1)
		tmp = Float64(1.0 + x);
	elseif (y <= 1.45e+18)
		tmp = fma(Float64(Float64(fma(0.16666666666666666, Float64(z * y), -0.5) * y) - z), y, Float64(1.0 + x));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.1], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 1.45e+18], N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision] + -0.5), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.0999999999999996 or 1.45e18 < 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. lower-+.f64 36.4

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

   5. Applied rewrites 36.4%

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

   if -5.0999999999999996 < y < 1.45e18

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 10: 69.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -46000000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -46000000000.0)
   (+ 1.0 x)
   (if (<= y 1.16e+44) (fma (- (* -0.5 y) z) y (+ 1.0 x)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -46000000000.0) {
		tmp = 1.0 + x;
	} else if (y <= 1.16e+44) {
		tmp = fma(((-0.5 * y) - z), y, (1.0 + x));
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -46000000000.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 1.16e+44)
		tmp = fma(Float64(Float64(-0.5 * y) - z), y, Float64(1.0 + x));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -46000000000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 1.16e+44], N[(N[(N[(-0.5 * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6e10 or 1.1600000000000001e44 < 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. lower-+.f64 35.9

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

   5. Applied rewrites 35.9%

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

   if -4.6e10 < y < 1.1600000000000001e44

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 94.6

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

   5. Applied rewrites 94.6%

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

Alternative 11: 69.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+52}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+43}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.8e+52)
   (+ 1.0 x)
   (if (<= y 8.4e+43) (- x (fma z y -1.0)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.8e+52) {
		tmp = 1.0 + x;
	} else if (y <= 8.4e+43) {
		tmp = x - fma(z, y, -1.0);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.8e+52)
		tmp = Float64(1.0 + x);
	elseif (y <= 8.4e+43)
		tmp = Float64(x - fma(z, y, -1.0));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.8e+52], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 8.4e+43], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.8e52 or 8.40000000000000007e43 < 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. lower-+.f64 35.8

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

   5. Applied rewrites 35.8%

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

   if -5.8e52 < y < 8.40000000000000007e43

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 90.6

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

   5. Applied rewrites 90.6%

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

Alternative 12: 61.6% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. lower-+.f64 62.6

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

5. Applied rewrites 62.6%

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

Alternative 13: 21.1% accurate, 212.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. lower-+.f64 62.6

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

5. Applied rewrites 62.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 21.0%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(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))))