Crypto.Random.Test:calculate from crypto-random-0.0.9

Percentage Accurate: 92.8% → 99.9%

Time: 5.7s

Alternatives: 4

Speedup: 1.1×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{y}{\frac{z}{y}} + x \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[x + \frac{y \cdot y}{z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \frac{y}{\frac{z}{y}} + x \]
6. Add Preprocessing

Alternative 2: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-58}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y y) 1e-58) (/ (* z x) z) (* (/ y z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * y) <= 1e-58) {
		tmp = (z * x) / z;
	} else {
		tmp = (y / 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 ((y * y) <= 1d-58) then
        tmp = (z * x) / z
    else
        tmp = (y / z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y * y) <= 1e-58) {
		tmp = (z * x) / z;
	} else {
		tmp = (y / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y * y) <= 1e-58:
		tmp = (z * x) / z
	else:
		tmp = (y / z) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * y) <= 1e-58)
		tmp = Float64(Float64(z * x) / z);
	else
		tmp = Float64(Float64(y / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y * y) <= 1e-58)
		tmp = (z * x) / z;
	else
		tmp = (y / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e-58], N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e-58

   1. Initial program 97.5%

      \[x + \frac{y \cdot y}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot z + {y}^{2}}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot z + {y}^{2}}{z}} \]

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in z around inf

      \[\leadsto \frac{x \cdot z}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.6%

      \[\leadsto \frac{x \cdot z}{z} \]

   if 1e-58 < (*.f64 y y)

   1. Initial program 91.5%

      \[x + \frac{y \cdot y}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{{y}^{2}}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{y}^{2}}{z}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 81.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-58}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[x + \frac{y \cdot y}{z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 4: 52.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[x + \frac{y \cdot y}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{{y}^{2}}{z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{y}^{2}}{z}} \]

   2. unpow2 N/A

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

   3. lower-*.f64 45.9

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

5. Applied rewrites 45.9%

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

7. Applied rewrites 51.2%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z)
  :name "Crypto.Random.Test:calculate from crypto-random-0.0.9"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (* y (/ y z))))

  (+ x (/ (* y y) z)))