SynthBasics:oscSampleBasedAux from YampaSynth-0.2

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 6

Speedup: 1.2×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return x + (y * (z - x))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return x + (y * (z - x))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fma (- z x) y x))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - x, y, x\right)} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z - x\right) \cdot y\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0105:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z x) y)))
   (if (<= y -9.2e+16) t_0 (if (<= y 0.0105) (fma z y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z - x) * y;
	double tmp;
	if (y <= -9.2e+16) {
		tmp = t_0;
	} else if (y <= 0.0105) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z - x) * y)
	tmp = 0.0
	if (y <= -9.2e+16)
		tmp = t_0;
	elseif (y <= 0.0105)
		tmp = fma(z, y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -9.2e+16], t$95$0, If[LessEqual[y, 0.0105], N[(z * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.2e16 or 0.0105000000000000007 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -9.2e16 < y < 0.0105000000000000007

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.7

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

   7. Applied rewrites 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+16}:\\ \;\;\;\;\left(z - x\right) \cdot y\\ \mathbf{elif}\;y \leq 0.0105:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z - x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.4e+229)
   (fma z y x)
   (if (<= y -3.7e+28) (* x (- y)) (fma z y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.4e+229) {
		tmp = fma(z, y, x);
	} else if (y <= -3.7e+28) {
		tmp = x * -y;
	} else {
		tmp = fma(z, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.4e+229)
		tmp = fma(z, y, x);
	elseif (y <= -3.7e+28)
		tmp = Float64(x * Float64(-y));
	else
		tmp = fma(z, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.4e+229], N[(z * y + x), $MachinePrecision], If[LessEqual[y, -3.7e+28], N[(x * (-y)), $MachinePrecision], N[(z * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.3999999999999995e229 or -3.6999999999999999e28 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if -8.3999999999999995e229 < y < -3.6999999999999999e28

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 67.0

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

   8. Applied rewrites 67.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 76.1

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

5. Applied rewrites 76.1%

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

   1. lift-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 76.1

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

7. Applied rewrites 76.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, x\right)} \]
8. Add Preprocessing

Alternative 5: 42.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ z \cdot y \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return z * 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 = z * y
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(z * y)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z * y), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 43.6

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

5. Applied rewrites 43.6%

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

6. Final simplification 43.6%

   \[\leadsto z \cdot y \]
7. Add Preprocessing

Alternative 6: 3.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(x * y)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * y), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 66.9

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

5. Applied rewrites 66.9%

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

6. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 27.6

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

8. Applied rewrites 27.6%

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

   1. neg-sub0 N/A

      \[\leadsto y \cdot \color{blue}{\left(0 - x\right)} \]

   2. flip3-- N/A

      \[\leadsto y \cdot \color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \]

   3. metadata-eval N/A

      \[\leadsto y \cdot \frac{\color{blue}{0} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   4. sub0-neg N/A

      \[\leadsto y \cdot \frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   5. sqr-pow N/A

      \[\leadsto y \cdot \frac{\mathsf{neg}\left(\color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   6. pow-prod-down N/A

      \[\leadsto y \cdot \frac{\mathsf{neg}\left(\color{blue}{{\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   7. sqr-neg N/A

      \[\leadsto y \cdot \frac{\mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   8. lift-neg.f64 N/A

      \[\leadsto y \cdot \frac{\mathsf{neg}\left({\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   9. lift-neg.f64 N/A

      \[\leadsto y \cdot \frac{\mathsf{neg}\left({\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\frac{3}{2}\right)}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   10. pow-prod-down N/A

       \[\leadsto y \cdot \frac{\mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   11. sqr-pow N/A

       \[\leadsto y \cdot \frac{\mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   12. lift-neg.f64 N/A

       \[\leadsto y \cdot \frac{\mathsf{neg}\left({\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   13. cube-neg N/A

       \[\leadsto y \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{3}\right)\right)}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   14. sub0-neg N/A

       \[\leadsto y \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(0 - {x}^{3}\right)}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   15. metadata-eval N/A

       \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(\color{blue}{{0}^{3}} - {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right)} \]

   17. flip3-- N/A

       \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(0 - x\right)}\right)\right) \]

   18. neg-sub0 N/A

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

   19. remove-double-neg N/A

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

10. Applied rewrites 5.6%

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

11. Final simplification 5.6%

    \[\leadsto x \cdot y \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (x y z)
  :name "SynthBasics:oscSampleBasedAux from YampaSynth-0.2"
  :precision binary64
  (+ x (* y (- z x))))