SynthBasics:oscSampleBasedAux from YampaSynth-0.2

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 5

Speedup: 1.2×

• 20.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. mul-1-neg N/A

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

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

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. associate-+l+ N/A

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-out-- N/A

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

   13. lower-fma.f64 N/A

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

   14. lower--.f64 100.0

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

5. Simplified 100.0%

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

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(z - x\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- z x))))
   (if (<= y -1.0) t_0 (if (<= y 2.35e-12) (fma z y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (z - x);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 2.35e-12) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(z - x))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 2.35e-12)
		tmp = fma(z, y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 2.35e-12], N[(z * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.34999999999999988e-12 < 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 98.0

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

   5. Simplified 98.0%

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

   if -1 < y < 2.34999999999999988e-12

   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 98.6

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

   5. Simplified 98.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 98.6

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

   8. Simplified 98.6%

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

Alternative 3: 75.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -y \cdot x\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+253}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y x))))
   (if (<= y -3.5e+253) t_0 (if (<= y 1.2e+194) (fma z y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(y * x);
	double tmp;
	if (y <= -3.5e+253) {
		tmp = t_0;
	} else if (y <= 1.2e+194) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(y * x))
	tmp = 0.0
	if (y <= -3.5e+253)
		tmp = t_0;
	elseif (y <= 1.2e+194)
		tmp = fma(z, y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(y * x), $MachinePrecision])}, If[LessEqual[y, -3.5e+253], t$95$0, If[LessEqual[y, 1.2e+194], N[(z * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.49999999999999978e253 or 1.2e194 < 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. Simplified 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 76.4

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

   8. Simplified 76.4%

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

   if -3.49999999999999978e253 < y < 1.2e194

   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 80.9

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

   5. Simplified 80.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 80.9

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

   8. Simplified 80.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+253}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;-y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% 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 75.1

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

5. Simplified 75.1%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 75.1

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

8. Simplified 75.1%

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

Alternative 5: 41.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * z), $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 36.0

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

5. Simplified 36.0%

   \[\leadsto \color{blue}{y \cdot z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (x y z)
  :name "SynthBasics:oscSampleBasedAux from YampaSynth-0.2"
  :precision binary64
  (+ x (* y (- z x))))