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

Percentage Accurate: 93.1% → 99.9%

Time: 3.1s

Alternatives: 4

Speedup: 1.0×

• 24.5% 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: 93.1% 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, 1.0× 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]

Derivation

1. Initial program 92.2%

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

   1. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* y y) z)))
   (if (<= t_0 -5e-25) (* y (/ y z)) (if (<= t_0 2e-50) x (/ y (/ z y))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * y) / z;
	double tmp;
	if (t_0 <= -5e-25) {
		tmp = y * (y / z);
	} else if (t_0 <= 2e-50) {
		tmp = x;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y * y) / z
    if (t_0 <= (-5d-25)) then
        tmp = y * (y / z)
    else if (t_0 <= 2d-50) then
        tmp = x
    else
        tmp = y / (z / y)
    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 <= -5e-25) {
		tmp = y * (y / z);
	} else if (t_0 <= 2e-50) {
		tmp = x;
	} else {
		tmp = y / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * y) / z
	tmp = 0
	if t_0 <= -5e-25:
		tmp = y * (y / z)
	elif t_0 <= 2e-50:
		tmp = x
	else:
		tmp = y / (z / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * y) / z)
	tmp = 0.0
	if (t_0 <= -5e-25)
		tmp = Float64(y * Float64(y / z));
	elseif (t_0 <= 2e-50)
		tmp = x;
	else
		tmp = Float64(y / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * y) / z;
	tmp = 0.0;
	if (t_0 <= -5e-25)
		tmp = y * (y / z);
	elseif (t_0 <= 2e-50)
		tmp = x;
	else
		tmp = y / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-25], N[(y * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-50], x, N[(y / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y y) z) < -4.99999999999999962e-25

   1. Initial program 84.7%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*l/ 84.7%

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

      3. *-un-lft-identity 84.7%

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

      4. add-sqr-sqrt 0.0%

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

      5. times-frac 0.0%

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

      6. fma-def 0.0%

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

      7. pow1/2 0.0%

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

      8. pow-flip 0.0%

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

      9. metadata-eval 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. associate-/l* 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in y around inf 77.6%

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

      1. unpow2 77.6%

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

   10. Simplified 77.6%

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

       1. associate-/l* 86.3%

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

       2. associate-/r/ 86.4%

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

   12. Applied egg-rr 86.4%

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

   if -4.99999999999999962e-25 < (/.f64 (*.f64 y y) z) < 2.00000000000000002e-50

   1. Initial program 99.2%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 91.0%

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

   if 2.00000000000000002e-50 < (/.f64 (*.f64 y y) z)

   1. Initial program 88.2%

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

      1. associate-*l/ 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*l/ 88.2%

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

      3. *-un-lft-identity 88.2%

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

      4. add-sqr-sqrt 88.0%

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

      5. times-frac 88.0%

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

      6. fma-def 88.0%

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

      7. pow1/2 88.0%

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

      8. pow-flip 88.0%

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

      9. metadata-eval 88.0%

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

   5. Applied egg-rr 88.0%

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

      1. associate-/l* 95.3%

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

   7. Simplified 95.3%

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

   8. Taylor expanded in y around inf 83.6%

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

      1. unpow2 83.6%

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

      2. associate-/l* 92.4%

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

   10. Simplified 92.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot y}{z} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y \cdot y}{z} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{y}}\\ \end{array} \]

Alternative 3: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-25} \lor \neg \left(y \leq 5.8 \cdot 10^{-20}\right):\\ \;\;\;\;y \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.1e-25) (not (<= y 5.8e-20))) (* y (/ y z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.1e-25) || !(y <= 5.8e-20)) {
		tmp = y * (y / z);
	} else {
		tmp = x;
	}
	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 <= (-1.1d-25)) .or. (.not. (y <= 5.8d-20))) then
        tmp = y * (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.1e-25) || !(y <= 5.8e-20)) {
		tmp = y * (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.1e-25) or not (y <= 5.8e-20):
		tmp = y * (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.1e-25) || !(y <= 5.8e-20))
		tmp = Float64(y * Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.1e-25) || ~((y <= 5.8e-20)))
		tmp = y * (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.1e-25], N[Not[LessEqual[y, 5.8e-20]], $MachinePrecision]], N[(y * N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.1000000000000001e-25 or 5.8e-20 < y

   1. Initial program 86.2%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*l/ 86.2%

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

      3. *-un-lft-identity 86.2%

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

      4. add-sqr-sqrt 40.1%

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

      5. times-frac 40.1%

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

      6. fma-def 40.1%

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

      7. pow1/2 40.1%

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

      8. pow-flip 40.1%

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

      9. metadata-eval 40.1%

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

   5. Applied egg-rr 40.1%

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

      1. associate-/l* 43.5%

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

   7. Simplified 43.5%

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

   8. Taylor expanded in y around inf 76.1%

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

      1. unpow2 76.1%

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

   10. Simplified 76.1%

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

       1. associate-/l* 85.0%

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

       2. associate-/r/ 85.0%

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

   12. Applied egg-rr 85.0%

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

   if -1.1000000000000001e-25 < y < 5.8e-20

   1. Initial program 99.2%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 84.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-25} \lor \neg \left(y \leq 5.8 \cdot 10^{-20}\right):\\ \;\;\;\;y \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 50.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 92.2%

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

   1. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 47.5%

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

5. Final simplification 47.5%

   \[\leadsto x \]

Developer target: 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 2023173 
(FPCore (x y z)
  :name "Crypto.Random.Test:calculate from crypto-random-0.0.9"
  :precision binary64

  :herbie-target
  (+ x (* y (/ y z)))

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