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

Percentage Accurate: 99.9% → 99.9%

Time: 9.3s

Alternatives: 16

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: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \mathsf{fma}\left(z, y, -1\right)\\ t_1 := \left(x + \cos y\right) - z \cdot \sin y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+128}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;t\_1 \leq -5000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.986:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (fma z y -1.0))) (t_1 (- (+ x (cos y)) (* z (sin y)))))
   (if (<= t_1 -5e+128)
     (+ 1.0 x)
     (if (<= t_1 -5000000.0) t_0 (if (<= t_1 0.986) (cos y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x - fma(z, y, -1.0);
	double t_1 = (x + cos(y)) - (z * sin(y));
	double tmp;
	if (t_1 <= -5e+128) {
		tmp = 1.0 + x;
	} else if (t_1 <= -5000000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.986) {
		tmp = cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x - fma(z, y, -1.0))
	t_1 = Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
	tmp = 0.0
	if (t_1 <= -5e+128)
		tmp = Float64(1.0 + x);
	elseif (t_1 <= -5000000.0)
		tmp = t_0;
	elseif (t_1 <= 0.986)
		tmp = cos(y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+128], N[(1.0 + x), $MachinePrecision], If[LessEqual[t$95$1, -5000000.0], t$95$0, If[LessEqual[t$95$1, 0.986], N[Cos[y], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -5e128

   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 70.2

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

   5. Applied rewrites 70.2%

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

   if -5e128 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -5e6 or 0.98599999999999999 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + -1 \cdot \left(y \cdot z\right)\right)} \]
   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 70.2

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

   5. Applied rewrites 70.2%

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

   if -5e6 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.98599999999999999

   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. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   6. 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 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \cos y \]
   9. Step-by-step derivation

   10. Applied rewrites 96.6%

       \[\leadsto \cos y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-10}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;x \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(-z, \sin y, \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e-10)
   (- (+ 1.0 x) (* z (sin y)))
   (if (<= x 0.00062)
     (fma (- z) (sin y) (cos y))
     (fma (sin y) (- z) (+ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-10) {
		tmp = (1.0 + x) - (z * sin(y));
	} else if (x <= 0.00062) {
		tmp = fma(-z, sin(y), cos(y));
	} else {
		tmp = fma(sin(y), -z, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e-10)
		tmp = Float64(Float64(1.0 + x) - Float64(z * sin(y)));
	elseif (x <= 0.00062)
		tmp = fma(Float64(-z), sin(y), cos(y));
	else
		tmp = fma(sin(y), Float64(-z), Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e-10], N[(N[(1.0 + x), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00062], N[((-z) * N[Sin[y], $MachinePrecision] + N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.30000000000000007e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.7%

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

   if -2.30000000000000007e-10 < x < 6.2e-4

   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. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 6.2e-4 < 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

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

   5. Applied rewrites 98.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 98.8

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 98.8

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

   7. Applied rewrites 98.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-10}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;x \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(-z, \sin y, \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \end{array} \]
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: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -600000000000:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 2.1:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -600000000000.0)
   (- (+ 1.0 x) (* z (sin y)))
   (if (<= z 2.1) (+ x (cos y)) (fma (sin y) (- z) (+ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -600000000000.0) {
		tmp = (1.0 + x) - (z * sin(y));
	} else if (z <= 2.1) {
		tmp = x + cos(y);
	} else {
		tmp = fma(sin(y), -z, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -600000000000.0)
		tmp = Float64(Float64(1.0 + x) - Float64(z * sin(y)));
	elseif (z <= 2.1)
		tmp = Float64(x + cos(y));
	else
		tmp = fma(sin(y), Float64(-z), Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -600000000000.0], N[(N[(1.0 + x), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6e11

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.7%

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

   if -6e11 < z < 2.10000000000000009

   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 99.7

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

   5. Applied rewrites 99.7%

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

   if 2.10000000000000009 < 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 98.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 98.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 98.2

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

   7. Applied rewrites 98.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -600000000000:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 2.1:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + x\right) - z \cdot \sin y\\ \mathbf{if}\;z \leq -600000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.1:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ 1.0 x) (* z (sin y)))))
   (if (<= z -600000000000.0) t_0 (if (<= z 2.1) (+ 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 <= -600000000000.0) {
		tmp = t_0;
	} else if (z <= 2.1) {
		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 <= (-600000000000.0d0)) then
        tmp = t_0
    else if (z <= 2.1d0) 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 <= -600000000000.0) {
		tmp = t_0;
	} else if (z <= 2.1) {
		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 <= -600000000000.0:
		tmp = t_0
	elif z <= 2.1:
		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 <= -600000000000.0)
		tmp = t_0;
	elseif (z <= 2.1)
		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 <= -600000000000.0)
		tmp = t_0;
	elseif (z <= 2.1)
		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, -600000000000.0], t$95$0, If[LessEqual[z, 2.1], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6e11 or 2.10000000000000009 < 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 98.9%

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

   if -6e11 < z < 2.10000000000000009

   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 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -600000000000:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 2.1:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-z, \sin y, 1\right)\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+110}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- z) (sin y) 1.0)))
   (if (<= z -1.95e+19) t_0 (if (<= z 1.32e+110) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-z, sin(y), 1.0);
	double tmp;
	if (z <= -1.95e+19) {
		tmp = t_0;
	} else if (z <= 1.32e+110) {
		tmp = x + cos(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-z), sin(y), 1.0)
	tmp = 0.0
	if (z <= -1.95e+19)
		tmp = t_0;
	elseif (z <= 1.32e+110)
		tmp = Float64(x + cos(y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[Sin[y], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[z, -1.95e+19], t$95$0, If[LessEqual[z, 1.32e+110], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.95e19 or 1.32e110 < 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. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 73.0

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-z, \sin y, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.0%

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

   if -1.95e19 < z < 1.32e110

   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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-z, \sin y, 1\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+110}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \sin y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+110}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))))
   (if (<= z -1.95e+19) t_0 (if (<= z 2.4e+110) (+ x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double tmp;
	if (z <= -1.95e+19) {
		tmp = t_0;
	} else if (z <= 2.4e+110) {
		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 <= (-1.95d+19)) then
        tmp = t_0
    else if (z <= 2.4d+110) 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 <= -1.95e+19) {
		tmp = t_0;
	} else if (z <= 2.4e+110) {
		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 <= -1.95e+19:
		tmp = t_0
	elif z <= 2.4e+110:
		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 <= -1.95e+19)
		tmp = t_0;
	elseif (z <= 2.4e+110)
		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 <= -1.95e+19)
		tmp = t_0;
	elseif (z <= 2.4e+110)
		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, -1.95e+19], t$95$0, If[LessEqual[z, 2.4e+110], N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.95e19 or 2.40000000000000012e110 < z

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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 64.1

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

   5. Applied rewrites 64.1%

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

   if -1.95e19 < z < 2.40000000000000012e110

   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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+19}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+110}:\\ \;\;\;\;x + \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ \mathbf{if}\;y \leq -30000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 25500:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right) \cdot z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))))
   (if (<= y -30000000.0)
     t_0
     (if (<= y 25500.0)
       (fma (* (fma 0.16666666666666666 (* y y) -1.0) 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 <= -30000000.0) {
		tmp = t_0;
	} else if (y <= 25500.0) {
		tmp = fma((fma(0.16666666666666666, (y * y), -1.0) * 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 <= -30000000.0)
		tmp = t_0;
	elseif (y <= 25500.0)
		tmp = fma(Float64(fma(0.16666666666666666, Float64(y * y), -1.0) * 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, -30000000.0], t$95$0, If[LessEqual[y, 25500.0], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision] * z), $MachinePrecision] * y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3e7 or 25500 < 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 58.7

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

   5. Applied rewrites 58.7%

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

   if -3e7 < y < 25500

   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%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right), y, 1 + x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -30000000:\\ \;\;\;\;x + \cos y\\ \mathbf{elif}\;y \leq 25500:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right) \cdot z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -220000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z \cdot y, -0.5\right) \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -220000000.0)
   (+ 1.0 x)
   (if (<= y 4.5e+14)
     (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 <= -220000000.0) {
		tmp = 1.0 + x;
	} else if (y <= 4.5e+14) {
		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 <= -220000000.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 4.5e+14)
		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, -220000000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 4.5e+14], 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 < -2.2e8 or 4.5e14 < 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 36.8

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

   5. Applied rewrites 36.8%

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

   if -2.2e8 < y < 4.5e14

   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 96.7

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

   5. Applied rewrites 96.7%

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

Alternative 11: 70.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -250000000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 980000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y \cdot y, -1\right) \cdot z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -250000000.0)
   (+ 1.0 x)
   (if (<= y 980000.0)
     (fma (* (fma 0.16666666666666666 (* y y) -1.0) z) y (+ 1.0 x))
     (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -250000000.0) {
		tmp = 1.0 + x;
	} else if (y <= 980000.0) {
		tmp = fma((fma(0.16666666666666666, (y * y), -1.0) * z), y, (1.0 + x));
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -250000000.0)
		tmp = Float64(1.0 + x);
	elseif (y <= 980000.0)
		tmp = fma(Float64(fma(0.16666666666666666, Float64(y * y), -1.0) * z), y, Float64(1.0 + x));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -250000000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 980000.0], N[(N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + -1.0), $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 < -2.5e8 or 9.8e5 < 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 36.5

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

   5. Applied rewrites 36.5%

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

   if -2.5e8 < y < 9.8e5

   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%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right), y, 1 + x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.3%

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

Alternative 12: 70.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+29}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, 1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.8e+29)
   (+ 1.0 x)
   (if (<= y 4.5e+15) (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 <= -3.8e+29) {
		tmp = 1.0 + x;
	} else if (y <= 4.5e+15) {
		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 <= -3.8e+29)
		tmp = Float64(1.0 + x);
	elseif (y <= 4.5e+15)
		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, -3.8e+29], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 4.5e+15], 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 < -3.79999999999999971e29 or 4.5e15 < 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 37.4

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

   5. Applied rewrites 37.4%

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

   if -3.79999999999999971e29 < y < 4.5e15

   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 93.7

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

   5. Applied rewrites 93.7%

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

Alternative 13: 70.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 3.15:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.4e+55)
   (+ 1.0 x)
   (if (<= y 3.15) (- x (fma z y -1.0)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.4e+55) {
		tmp = 1.0 + x;
	} else if (y <= 3.15) {
		tmp = x - fma(z, y, -1.0);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.4e+55)
		tmp = Float64(1.0 + x);
	elseif (y <= 3.15)
		tmp = Float64(x - fma(z, y, -1.0));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.4e+55], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 3.15], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.40000000000000021e55 or 3.14999999999999991 < 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 37.5

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

   5. Applied rewrites 37.5%

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

   if -4.40000000000000021e55 < y < 3.14999999999999991

   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 93.1

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

   5. Applied rewrites 93.1%

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

Alternative 14: 67.4% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -98000:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\mathsf{fma}\left(-y, z, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -98000.0) (+ 1.0 x) (if (<= x 4.0) (fma (- y) z 1.0) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -98000.0) {
		tmp = 1.0 + x;
	} else if (x <= 4.0) {
		tmp = fma(-y, z, 1.0);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -98000.0)
		tmp = Float64(1.0 + x);
	elseif (x <= 4.0)
		tmp = fma(Float64(-y), z, 1.0);
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -98000.0], N[(1.0 + x), $MachinePrecision], If[LessEqual[x, 4.0], N[((-y) * z + 1.0), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -98000 or 4 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if -98000 < x < 4

   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. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 82.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + -1 \cdot \left(y \cdot z\right)\right)} \]
   9. 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 48.1

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

   10. Applied rewrites 48.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 47.6%

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

Alternative 15: 62.2% 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 59.2

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

5. Applied rewrites 59.2%

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

Alternative 16: 21.4% 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 59.2

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

5. Applied rewrites 59.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.7%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024243 
(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))))