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

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

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 100.0

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

   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 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \cos y\\ t_1 := z \cdot \sin y\\ t_2 := t\_0 - t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \mathbf{elif}\;t\_2 \leq 0.9999999999999775:\\ \;\;\;\;\frac{1}{\frac{1}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (cos y))) (t_1 (* z (sin y))) (t_2 (- t_0 t_1)))
   (if (<= t_2 -1e+17)
     (fma (sin y) (- z) (+ 1.0 x))
     (if (<= t_2 0.9999999999999775) (/ 1.0 (/ 1.0 t_0)) (- (+ 1.0 x) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = x + cos(y);
	double t_1 = z * sin(y);
	double t_2 = t_0 - t_1;
	double tmp;
	if (t_2 <= -1e+17) {
		tmp = fma(sin(y), -z, (1.0 + x));
	} else if (t_2 <= 0.9999999999999775) {
		tmp = 1.0 / (1.0 / t_0);
	} else {
		tmp = (1.0 + x) - t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + cos(y))
	t_1 = Float64(z * sin(y))
	t_2 = Float64(t_0 - t_1)
	tmp = 0.0
	if (t_2 <= -1e+17)
		tmp = fma(sin(y), Float64(-z), Float64(1.0 + x));
	elseif (t_2 <= 0.9999999999999775)
		tmp = Float64(1.0 / Float64(1.0 / t_0));
	else
		tmp = Float64(Float64(1.0 + x) - t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+17], N[(N[Sin[y], $MachinePrecision] * (-z) + N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999999999999775], N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -1e17

   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 99.9%

      \[\leadsto \left(x + \color{blue}{1}\right) - z \cdot \sin y \]
   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-sub-sign-inv N/A

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

      4. lift-sin.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -1e17 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.999999999999977462

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   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. lower-/.f64 99.5

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x + \cos y}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x + \cos y}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\cos y + x}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\cos y + x}}} \]

      4. lower-cos.f64 99.0

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\cos y} + x}} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\cos y + x}}} \]

   if 0.999999999999977462 < (-.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

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + \cos y\right) - z \cdot \sin y \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \mathbf{elif}\;\left(x + \cos y\right) - z \cdot \sin y \leq 0.9999999999999775:\\ \;\;\;\;\frac{1}{\frac{1}{x + \cos y}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 92.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-151}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-47}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.3e-151)
   (- (+ 1.0 x) (* z (sin y)))
   (if (<= z 2.6e-47)
     (- (+ x (cos y)) (* z y))
     (fma (sin y) (- z) (+ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e-151) {
		tmp = (1.0 + x) - (z * sin(y));
	} else if (z <= 2.6e-47) {
		tmp = (x + cos(y)) - (z * y);
	} else {
		tmp = fma(sin(y), -z, (1.0 + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.3e-151)
		tmp = Float64(Float64(1.0 + x) - Float64(z * sin(y)));
	elseif (z <= 2.6e-47)
		tmp = Float64(Float64(x + cos(y)) - Float64(z * y));
	else
		tmp = fma(sin(y), Float64(-z), Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.3e-151], N[(N[(1.0 + x), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e-47], N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * 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 < -1.3e-151

   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 93.8%

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

   if -1.3e-151 < z < 2.6e-47

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 2.6e-47 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.5%

      \[\leadsto \left(x + \color{blue}{1}\right) - z \cdot \sin y \]
   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-sub-sign-inv N/A

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

      4. lift-sin.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 97.5

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.5

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

   7. Applied rewrites 97.5%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-151}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-47}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, 1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + x\right) - z \cdot \sin y\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{-151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-47}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ 1.0 x) (* z (sin y)))))
   (if (<= z -1.3e-151)
     t_0
     (if (<= z 2.6e-47) (- (+ x (cos y)) (* z 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 <= -1.3e-151) {
		tmp = t_0;
	} else if (z <= 2.6e-47) {
		tmp = (x + cos(y)) - (z * 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 <= (-1.3d-151)) then
        tmp = t_0
    else if (z <= 2.6d-47) then
        tmp = (x + cos(y)) - (z * 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 <= -1.3e-151) {
		tmp = t_0;
	} else if (z <= 2.6e-47) {
		tmp = (x + Math.cos(y)) - (z * 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 <= -1.3e-151:
		tmp = t_0
	elif z <= 2.6e-47:
		tmp = (x + math.cos(y)) - (z * 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 <= -1.3e-151)
		tmp = t_0;
	elseif (z <= 2.6e-47)
		tmp = Float64(Float64(x + cos(y)) - Float64(z * 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 <= -1.3e-151)
		tmp = t_0;
	elseif (z <= 2.6e-47)
		tmp = (x + cos(y)) - (z * 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, -1.3e-151], t$95$0, If[LessEqual[z, 2.6e-47], N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e-151 or 2.6e-47 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 95.5%

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

   if -1.3e-151 < z < 2.6e-47

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-151}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-47}:\\ \;\;\;\;\left(x + \cos y\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - z \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \sin y\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;x - \mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(\frac{\frac{y}{z} \cdot 0.25}{z \cdot z}, y, \frac{1}{z}\right)}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (sin y))))
   (if (<= z -7.5e+148)
     t_0
     (if (<= z 4.8e+70)
       (-
        x
        (fma y (/ 1.0 (fma (/ (* (/ y z) 0.25) (* z z)) y (/ 1.0 z))) -1.0))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * sin(y);
	double tmp;
	if (z <= -7.5e+148) {
		tmp = t_0;
	} else if (z <= 4.8e+70) {
		tmp = x - fma(y, (1.0 / fma((((y / z) * 0.25) / (z * z)), y, (1.0 / z))), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * sin(y))
	tmp = 0.0
	if (z <= -7.5e+148)
		tmp = t_0;
	elseif (z <= 4.8e+70)
		tmp = Float64(x - fma(y, Float64(1.0 / fma(Float64(Float64(Float64(y / z) * 0.25) / Float64(z * z)), y, Float64(1.0 / z))), -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.5e+148], t$95$0, If[LessEqual[z, 4.8e+70], N[(x - N[(y * N[(1.0 / N[(N[(N[(N[(y / z), $MachinePrecision] * 0.25), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] * y + N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000008e148 or 4.79999999999999974e70 < 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 68.5

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

   5. Applied rewrites 68.5%

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

   if -7.50000000000000008e148 < z < 4.79999999999999974e70

   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. +-commutative N/A

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

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

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

   5. Applied rewrites 62.8%

      \[\leadsto \color{blue}{x - \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y, z\right), -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto x - \mathsf{fma}\left(y, \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, y, -z\right)}{\mathsf{fma}\left(0.25, y \cdot y, \left(-z\right) \cdot z\right)}}}, -1\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto x - \mathsf{fma}\left(y, \frac{1}{y \cdot \left(\frac{1}{4} \cdot \frac{y}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{{z}^{2}}\right) + \color{blue}{\frac{1}{z}}}, -1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.4%

       \[\leadsto x - \mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(\frac{-0.5 + \frac{0.25 \cdot y}{z}}{z \cdot z}, \color{blue}{y}, \frac{1}{z}\right)}, -1\right) \]

   11. Taylor expanded in y around inf

       \[\leadsto x - \mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(\frac{\frac{1}{4} \cdot \frac{y}{z}}{z \cdot z}, y, \frac{1}{z}\right)}, -1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 78.4%

       \[\leadsto x - \mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(\frac{\frac{y}{z} \cdot 0.25}{z \cdot z}, y, \frac{1}{z}\right)}, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 88.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + x\right) - z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ 1.0 x) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (1.0 + x) - (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 = (1.0d0 + x) - (z * sin(y))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 + x) - (z * Math.sin(y));
}

Python

def code(x, y, z):
	return (1.0 + x) - (z * math.sin(y))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(1.0 + x), $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. 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 86.9%

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

6. Final simplification 86.9%

   \[\leadsto \left(1 + x\right) - z \cdot \sin y \]
7. Add Preprocessing

Alternative 8: 68.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x - \mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(\frac{\frac{y}{z} \cdot 0.25}{z \cdot z}, y, \frac{1}{z}\right)}, -1\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- x (fma y (/ 1.0 (fma (/ (* (/ y z) 0.25) (* z z)) y (/ 1.0 z))) -1.0)))

C

double code(double x, double y, double z) {
	return x - fma(y, (1.0 / fma((((y / z) * 0.25) / (z * z)), y, (1.0 / z))), -1.0);
}

Julia

function code(x, y, z)
	return Float64(x - fma(y, Float64(1.0 / fma(Float64(Float64(Float64(y / z) * 0.25) / Float64(z * z)), y, Float64(1.0 / z))), -1.0))
end

Wolfram

code[x_, y_, z_] := N[(x - N[(y * N[(1.0 / N[(N[(N[(N[(y / z), $MachinePrecision] * 0.25), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] * y + N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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

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

   1. +-commutative N/A

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

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

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

   3. associate-+l- N/A

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

   4. lower--.f64 N/A

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

   5. sub-neg N/A

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

5. Applied rewrites 58.6%

   \[\leadsto \color{blue}{x - \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y, z\right), -1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 48.2%

   \[\leadsto x - \mathsf{fma}\left(y, \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, y, -z\right)}{\mathsf{fma}\left(0.25, y \cdot y, \left(-z\right) \cdot z\right)}}}, -1\right) \]

8. Taylor expanded in y around 0

   \[\leadsto x - \mathsf{fma}\left(y, \frac{1}{y \cdot \left(\frac{1}{4} \cdot \frac{y}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{{z}^{2}}\right) + \color{blue}{\frac{1}{z}}}, -1\right) \]
9. Step-by-step derivation

10. Applied rewrites 69.9%

    \[\leadsto x - \mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(\frac{-0.5 + \frac{0.25 \cdot y}{z}}{z \cdot z}, \color{blue}{y}, \frac{1}{z}\right)}, -1\right) \]

11. Taylor expanded in y around inf

    \[\leadsto x - \mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(\frac{\frac{1}{4} \cdot \frac{y}{z}}{z \cdot z}, y, \frac{1}{z}\right)}, -1\right) \]
12. Step-by-step derivation

13. Applied rewrites 69.9%

    \[\leadsto x - \mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(\frac{\frac{y}{z} \cdot 0.25}{z \cdot z}, y, \frac{1}{z}\right)}, -1\right) \]
14. Add Preprocessing

Alternative 9: 69.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{+23}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 10^{+18}:\\ \;\;\;\;x - \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y, z\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.95e+23)
   (+ 1.0 x)
   (if (<= y 1e+18) (- x (fma y (fma 0.5 y z) -1.0)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.95e+23) {
		tmp = 1.0 + x;
	} else if (y <= 1e+18) {
		tmp = x - fma(y, fma(0.5, y, z), -1.0);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.95e+23)
		tmp = Float64(1.0 + x);
	elseif (y <= 1e+18)
		tmp = Float64(x - fma(y, fma(0.5, y, z), -1.0));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.95e+23], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 1e+18], N[(x - N[(y * N[(0.5 * y + z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.94999999999999994e23 or 1e18 < 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

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

      1. lower-+.f64 42.6

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

   5. Applied rewrites 42.6%

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

   if -2.94999999999999994e23 < y < 1e18

   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. +-commutative N/A

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

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

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

   5. Applied rewrites 96.1%

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

Alternative 10: 69.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+77}:\\ \;\;\;\;1 + x\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+18}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.15e+77)
   (+ 1.0 x)
   (if (<= y 1.25e+18) (- x (fma z y -1.0)) (+ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.15e+77) {
		tmp = 1.0 + x;
	} else if (y <= 1.25e+18) {
		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 <= -2.15e+77)
		tmp = Float64(1.0 + x);
	elseif (y <= 1.25e+18)
		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, -2.15e+77], N[(1.0 + x), $MachinePrecision], If[LessEqual[y, 1.25e+18], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.14999999999999996e77 or 1.25e18 < 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

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

      1. lower-+.f64 43.3

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

   5. Applied rewrites 43.3%

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

   if -2.14999999999999996e77 < y < 1.25e18

   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. distribute-rgt-neg-in N/A

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

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

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

Alternative 11: 63.9% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 10^{+170}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1e+170) (+ 1.0 x) (- x (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1e+170) {
		tmp = 1.0 + x;
	} else {
		tmp = x - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1d+170) then
        tmp = 1.0d0 + x
    else
        tmp = x - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1e+170) {
		tmp = 1.0 + x;
	} else {
		tmp = x - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1e+170:
		tmp = 1.0 + x
	else:
		tmp = x - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1e+170)
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(x - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1e+170)
		tmp = 1.0 + x;
	else
		tmp = x - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1e+170], N[(1.0 + x), $MachinePrecision], N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.00000000000000003e170

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if 1.00000000000000003e170 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 26.1%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 67.2%

       \[\leadsto x - y \cdot \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+170}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.6% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{+230}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.05e+230) (+ 1.0 x) (* (- y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.05e+230) {
		tmp = 1.0 + x;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.05d+230) then
        tmp = 1.0d0 + x
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.05e+230) {
		tmp = 1.0 + x;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.05e+230:
		tmp = 1.0 + x
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.05e+230)
		tmp = Float64(1.0 + x);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.05e+230)
		tmp = 1.0 + x;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.05e+230], N[(1.0 + x), $MachinePrecision], N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.04999999999999996e230

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if 1.04999999999999996e230 < 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 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 46.2%

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

Alternative 13: 61.8% 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 63.7

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

5. Applied rewrites 63.7%

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

Alternative 14: 22.4% 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 63.7

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

5. Applied rewrites 63.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 23.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(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))))