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

Percentage Accurate: 93.5% → 99.9%

Time: 5.4s

Alternatives: 3

Speedup: 1.1×

• 25.3% 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: 93.5% 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, 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 2: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot y}{z}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+154} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{y}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* y y) z)))
   (if (or (<= t_0 -4e+154) (not (<= t_0 2e-29)))
     (* (/ y z) y)
     (* (- x) -1.0))))

C

double code(double x, double y, double z) {
	double t_0 = (y * y) / z;
	double tmp;
	if ((t_0 <= -4e+154) || !(t_0 <= 2e-29)) {
		tmp = (y / z) * y;
	} else {
		tmp = -x * -1.0;
	}
	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 = (y * y) / z
    if ((t_0 <= (-4d+154)) .or. (.not. (t_0 <= 2d-29))) then
        tmp = (y / z) * y
    else
        tmp = -x * (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * y) / z;
	double tmp;
	if ((t_0 <= -4e+154) || !(t_0 <= 2e-29)) {
		tmp = (y / z) * y;
	} else {
		tmp = -x * -1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * y) / z
	tmp = 0
	if (t_0 <= -4e+154) or not (t_0 <= 2e-29):
		tmp = (y / z) * y
	else:
		tmp = -x * -1.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * y) / z)
	tmp = 0.0
	if ((t_0 <= -4e+154) || !(t_0 <= 2e-29))
		tmp = Float64(Float64(y / z) * y);
	else
		tmp = Float64(Float64(-x) * -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * y) / z;
	tmp = 0.0;
	if ((t_0 <= -4e+154) || ~((t_0 <= 2e-29)))
		tmp = (y / z) * y;
	else
		tmp = -x * -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e+154], N[Not[LessEqual[t$95$0, 2e-29]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * y), $MachinePrecision], N[((-x) * -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y y) z) < -4.00000000000000015e154 or 1.99999999999999989e-29 < (/.f64 (*.f64 y y) z)

   1. Initial program 90.5%

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

   3. Taylor expanded in x 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 85.3

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 94.6%

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

   if -4.00000000000000015e154 < (/.f64 (*.f64 y y) z) < 1.99999999999999989e-29

   1. Initial program 97.7%

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

   3. Taylor expanded in x 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 14.4

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

   5. Applied rewrites 14.4%

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

   7. Applied rewrites 16.7%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l/ N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-frac-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 97.6

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

   10. Applied rewrites 97.6%

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

   11. Taylor expanded in x around inf

       \[\leadsto \left(-x\right) \cdot -1 \]
   12. Step-by-step derivation

   13. Applied rewrites 86.3%

       \[\leadsto \left(-x\right) \cdot -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot y}{z} \leq -4 \cdot 10^{+154} \lor \neg \left(\frac{y \cdot y}{z} \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{y}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return -x * -1.0

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[((-x) * -1.0), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in x 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 50.1

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

5. Applied rewrites 50.1%

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

7. Applied rewrites 55.9%

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

8. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower--.f64 N/A

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

   6. mul-1-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l/ N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. distribute-neg-frac N/A

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

   13. lower-/.f64 N/A

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

   14. distribute-frac-neg N/A

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

   15. lower-/.f64 N/A

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

   16. lower-neg.f64 91.8

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

10. Applied rewrites 91.8%

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

11. Taylor expanded in x around inf

    \[\leadsto \left(-x\right) \cdot -1 \]
12. Step-by-step derivation

13. Applied rewrites 46.2%

    \[\leadsto \left(-x\right) \cdot -1 \]
14. 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 2024339 
(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)))