a parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.4%

Time: 6.7s

Alternatives: 9

Speedup: 1.0×

• 42.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * m
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * m

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * m;
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * m
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * m

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * m;
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.3e-32) (* (- (/ m v) 1.0) m) (/ (* (* (- 1.0 m) m) m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.3e-32) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = (((1.0 - m) * m) * m) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.3d-32) then
        tmp = ((m / v) - 1.0d0) * m
    else
        tmp = (((1.0d0 - m) * m) * m) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.3e-32) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = (((1.0 - m) * m) * m) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.3e-32:
		tmp = ((m / v) - 1.0) * m
	else:
		tmp = (((1.0 - m) * m) * m) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.3e-32)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * m);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) * m) * m) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.3e-32)
		tmp = ((m / v) - 1.0) * m;
	else
		tmp = (((1.0 - m) * m) * m) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.3e-32], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.3000000000000001e-32

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]
   4. Step-by-step derivation

      1. lower-/.f64 99.7

         \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]

   5. Applied rewrites 99.7%

      \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]

   if 2.3000000000000001e-32 < m

   1. Initial program 99.8%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right) - 1\right)} \cdot m \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(m \cdot \color{blue}{\left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)} - 1\right) \cdot m \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(m \cdot \color{blue}{\left(\frac{1}{v} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{m}{v}\right)} - 1\right) \cdot m \]

      3. metadata-eval N/A

         \[\leadsto \left(m \cdot \left(\frac{1}{v} - \color{blue}{1} \cdot \frac{m}{v}\right) - 1\right) \cdot m \]

      4. *-lft-identity N/A

         \[\leadsto \left(m \cdot \left(\frac{1}{v} - \color{blue}{\frac{m}{v}}\right) - 1\right) \cdot m \]

      5. div-sub N/A

         \[\leadsto \left(m \cdot \color{blue}{\frac{1 - m}{v}} - 1\right) \cdot m \]

      6. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} - 1\right) \cdot m \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \color{blue}{1 \cdot 1}\right) \cdot m \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot 1\right) \cdot m \]

      9. *-inverses N/A

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \color{blue}{\frac{v}{v}}\right) \cdot m \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + -1 \cdot \frac{v}{v}\right)} \cdot m \]

      11. *-inverses N/A

          \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} + -1 \cdot \color{blue}{1}\right) \cdot m \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} + \color{blue}{-1}\right) \cdot m \]

      13. associate-/l* N/A

          \[\leadsto \left(\color{blue}{m \cdot \frac{1 - m}{v}} + -1\right) \cdot m \]

      14. *-commutative N/A

          \[\leadsto \left(\color{blue}{\frac{1 - m}{v} \cdot m} + -1\right) \cdot m \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right)} \cdot m \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1 - m}{v}}, m, -1\right) \cdot m \]

      17. lower--.f64 99.8

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 - m}}{v}, m, -1\right) \cdot m \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right)} \cdot m \]

   6. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\left(\left(1 - m\right) \cdot m\right) \cdot m}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \leq -5 \cdot 10^{-306}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* m (- 1.0 m)) v) 1.0) m) -5e-306) (- m) (* (/ m v) m)))

C

double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -5e-306) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (((((m * (1.0d0 - m)) / v) - 1.0d0) * m) <= (-5d-306)) then
        tmp = -m
    else
        tmp = (m / v) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -5e-306) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if ((((m * (1.0 - m)) / v) - 1.0) * m) <= -5e-306:
		tmp = -m
	else:
		tmp = (m / v) * m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m) <= -5e-306)
		tmp = Float64(-m);
	else
		tmp = Float64(Float64(m / v) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -5e-306)
		tmp = -m;
	else
		tmp = (m / v) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -5e-306], (-m), N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -4.99999999999999998e-306

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{-1 \cdot m} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(m\right)} \]

      2. lower-neg.f64 33.0

         \[\leadsto \color{blue}{-m} \]

   5. Applied rewrites 33.0%

      \[\leadsto \color{blue}{-m} \]

   if -4.99999999999999998e-306 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.4%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} \cdot m \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} \cdot m \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 - m\right) \cdot m}}{v} \cdot m \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 - m\right) \cdot m}}{v} \cdot m \]

      4. lower--.f64 96.4

         \[\leadsto \frac{\color{blue}{\left(1 - m\right)} \cdot m}{v} \cdot m \]

   5. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\frac{\left(1 - m\right) \cdot m}{v}} \cdot m \]

   6. Taylor expanded in m around 0

      \[\leadsto \frac{m}{\color{blue}{v}} \cdot m \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{m}{\color{blue}{v}} \cdot m \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 43.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \leq -5 \cdot 10^{-306}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* m (- 1.0 m)) v) 1.0) m) -5e-306) (- m) (/ (* m m) v)))

C

double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -5e-306) {
		tmp = -m;
	} else {
		tmp = (m * m) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (((((m * (1.0d0 - m)) / v) - 1.0d0) * m) <= (-5d-306)) then
        tmp = -m
    else
        tmp = (m * m) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -5e-306) {
		tmp = -m;
	} else {
		tmp = (m * m) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if ((((m * (1.0 - m)) / v) - 1.0) * m) <= -5e-306:
		tmp = -m
	else:
		tmp = (m * m) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m) <= -5e-306)
		tmp = Float64(-m);
	else
		tmp = Float64(Float64(m * m) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -5e-306)
		tmp = -m;
	else
		tmp = (m * m) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -5e-306], (-m), N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -4.99999999999999998e-306

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{-1 \cdot m} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(m\right)} \]

      2. lower-neg.f64 33.0

         \[\leadsto \color{blue}{-m} \]

   5. Applied rewrites 33.0%

      \[\leadsto \color{blue}{-m} \]

   if -4.99999999999999998e-306 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.4%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right) - 1\right)} \cdot m \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(m \cdot \color{blue}{\left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)} - 1\right) \cdot m \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(m \cdot \color{blue}{\left(\frac{1}{v} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{m}{v}\right)} - 1\right) \cdot m \]

      3. metadata-eval N/A

         \[\leadsto \left(m \cdot \left(\frac{1}{v} - \color{blue}{1} \cdot \frac{m}{v}\right) - 1\right) \cdot m \]

      4. *-lft-identity N/A

         \[\leadsto \left(m \cdot \left(\frac{1}{v} - \color{blue}{\frac{m}{v}}\right) - 1\right) \cdot m \]

      5. div-sub N/A

         \[\leadsto \left(m \cdot \color{blue}{\frac{1 - m}{v}} - 1\right) \cdot m \]

      6. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} - 1\right) \cdot m \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \color{blue}{1 \cdot 1}\right) \cdot m \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot 1\right) \cdot m \]

      9. *-inverses N/A

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \color{blue}{\frac{v}{v}}\right) \cdot m \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + -1 \cdot \frac{v}{v}\right)} \cdot m \]

      11. *-inverses N/A

          \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} + -1 \cdot \color{blue}{1}\right) \cdot m \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} + \color{blue}{-1}\right) \cdot m \]

      13. associate-/l* N/A

          \[\leadsto \left(\color{blue}{m \cdot \frac{1 - m}{v}} + -1\right) \cdot m \]

      14. *-commutative N/A

          \[\leadsto \left(\color{blue}{\frac{1 - m}{v} \cdot m} + -1\right) \cdot m \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right)} \cdot m \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1 - m}{v}}, m, -1\right) \cdot m \]

      17. lower--.f64 99.4

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 - m}}{v}, m, -1\right) \cdot m \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right)} \cdot m \]

   6. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]

   8. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{\left(\left(1 - m\right) \cdot m\right) \cdot m}{v}} \]

   9. Taylor expanded in m around 0

      \[\leadsto \frac{{m}^{2}}{v} \]
   10. Step-by-step derivation

   11. Applied rewrites 75.1%

       \[\leadsto \frac{m \cdot m}{v} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.3e-32) (* (- (/ m v) 1.0) m) (* (- 1.0 m) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.3e-32) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = (1.0 - m) * (m * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.3d-32) then
        tmp = ((m / v) - 1.0d0) * m
    else
        tmp = (1.0d0 - m) * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.3e-32) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = (1.0 - m) * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.3e-32:
		tmp = ((m / v) - 1.0) * m
	else:
		tmp = (1.0 - m) * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.3e-32)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * m);
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.3e-32)
		tmp = ((m / v) - 1.0) * m;
	else
		tmp = (1.0 - m) * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.3e-32], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.3000000000000001e-32

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]
   4. Step-by-step derivation

      1. lower-/.f64 99.7

         \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]

   5. Applied rewrites 99.7%

      \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]

   if 2.3000000000000001e-32 < m

   1. Initial program 99.8%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right) - 1\right)} \cdot m \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(m \cdot \color{blue}{\left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)} - 1\right) \cdot m \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(m \cdot \color{blue}{\left(\frac{1}{v} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{m}{v}\right)} - 1\right) \cdot m \]

      3. metadata-eval N/A

         \[\leadsto \left(m \cdot \left(\frac{1}{v} - \color{blue}{1} \cdot \frac{m}{v}\right) - 1\right) \cdot m \]

      4. *-lft-identity N/A

         \[\leadsto \left(m \cdot \left(\frac{1}{v} - \color{blue}{\frac{m}{v}}\right) - 1\right) \cdot m \]

      5. div-sub N/A

         \[\leadsto \left(m \cdot \color{blue}{\frac{1 - m}{v}} - 1\right) \cdot m \]

      6. associate-/l* N/A

         \[\leadsto \left(\color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} - 1\right) \cdot m \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \color{blue}{1 \cdot 1}\right) \cdot m \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot 1\right) \cdot m \]

      9. *-inverses N/A

         \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \color{blue}{\frac{v}{v}}\right) \cdot m \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + -1 \cdot \frac{v}{v}\right)} \cdot m \]

      11. *-inverses N/A

          \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} + -1 \cdot \color{blue}{1}\right) \cdot m \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} + \color{blue}{-1}\right) \cdot m \]

      13. associate-/l* N/A

          \[\leadsto \left(\color{blue}{m \cdot \frac{1 - m}{v}} + -1\right) \cdot m \]

      14. *-commutative N/A

          \[\leadsto \left(\color{blue}{\frac{1 - m}{v} \cdot m} + -1\right) \cdot m \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right)} \cdot m \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1 - m}{v}}, m, -1\right) \cdot m \]

      17. lower--.f64 99.8

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 - m}}{v}, m, -1\right) \cdot m \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right)} \cdot m \]

   6. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\left(\left(1 - m\right) \cdot m\right) \cdot m}{v}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

       \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(m \cdot \frac{m}{v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot m}{-v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0) (* (- (/ m v) 1.0) m) (/ (* (* m m) m) (- v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = ((m * m) * m) / -v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.0d0) then
        tmp = ((m / v) - 1.0d0) * m
    else
        tmp = ((m * m) * m) / -v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = ((m * m) * m) / -v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = ((m / v) - 1.0) * m
	else:
		tmp = ((m * m) * m) / -v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * m);
	else
		tmp = Float64(Float64(Float64(m * m) * m) / Float64(-v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = ((m / v) - 1.0) * m;
	else
		tmp = ((m * m) * m) / -v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], N[(N[(N[(m * m), $MachinePrecision] * m), $MachinePrecision] / (-v)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]
   4. Step-by-step derivation

      1. lower-/.f64 97.4

         \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]

   5. Applied rewrites 97.4%

      \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {m}^{3}}{v}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {m}^{3}}{v}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{{-1}^{3}} \cdot {m}^{3}}{v} \]

      4. cube-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(-1 \cdot m\right)}^{3}}}{v} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(-1 \cdot m\right)}^{3}}}{v} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{neg}\left(m\right)\right)}}^{3}}{v} \]

      7. lower-neg.f64 97.3

         \[\leadsto \frac{{\color{blue}{\left(-m\right)}}^{3}}{v} \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\frac{{\left(-m\right)}^{3}}{v}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.3%

      \[\leadsto \frac{\left(m \cdot m\right) \cdot \left(-m\right)}{v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot m}{-v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * m
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * m

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * m;
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 7: 50.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.0) (* (- (/ m v) 1.0) m) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = -m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.0d0) then
        tmp = ((m / v) - 1.0d0) * m
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = ((m / v) - 1.0) * m
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * m);
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = ((m / v) - 1.0) * m;
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]
   4. Step-by-step derivation

      1. lower-/.f64 97.4

         \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]

   5. Applied rewrites 97.4%

      \[\leadsto \left(\color{blue}{\frac{m}{v}} - 1\right) \cdot m \]

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{-1 \cdot m} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(m\right)} \]

      2. lower-neg.f64 5.4

         \[\leadsto \color{blue}{-m} \]

   5. Applied rewrites 5.4%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right) \cdot m \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (fma (/ (- 1.0 m) v) m -1.0) m))

C

double code(double m, double v) {
	return fma(((1.0 - m) / v), m, -1.0) * m;
}

Julia

function code(m, v)
	return Float64(fma(Float64(Float64(1.0 - m) / v), m, -1.0) * m)
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision] * m + -1.0), $MachinePrecision] * m), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
2. Add Preprocessing

3. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right) - 1\right)} \cdot m \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(m \cdot \color{blue}{\left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)} - 1\right) \cdot m \]

   2. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(m \cdot \color{blue}{\left(\frac{1}{v} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{m}{v}\right)} - 1\right) \cdot m \]

   3. metadata-eval N/A

      \[\leadsto \left(m \cdot \left(\frac{1}{v} - \color{blue}{1} \cdot \frac{m}{v}\right) - 1\right) \cdot m \]

   4. *-lft-identity N/A

      \[\leadsto \left(m \cdot \left(\frac{1}{v} - \color{blue}{\frac{m}{v}}\right) - 1\right) \cdot m \]

   5. div-sub N/A

      \[\leadsto \left(m \cdot \color{blue}{\frac{1 - m}{v}} - 1\right) \cdot m \]

   6. associate-/l* N/A

      \[\leadsto \left(\color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} - 1\right) \cdot m \]

   7. metadata-eval N/A

      \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \color{blue}{1 \cdot 1}\right) \cdot m \]

   8. metadata-eval N/A

      \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot 1\right) \cdot m \]

   9. *-inverses N/A

      \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \color{blue}{\frac{v}{v}}\right) \cdot m \]

   10. fp-cancel-sign-sub-inv N/A

       \[\leadsto \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + -1 \cdot \frac{v}{v}\right)} \cdot m \]

   11. *-inverses N/A

       \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} + -1 \cdot \color{blue}{1}\right) \cdot m \]

   12. metadata-eval N/A

       \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} + \color{blue}{-1}\right) \cdot m \]

   13. associate-/l* N/A

       \[\leadsto \left(\color{blue}{m \cdot \frac{1 - m}{v}} + -1\right) \cdot m \]

   14. *-commutative N/A

       \[\leadsto \left(\color{blue}{\frac{1 - m}{v} \cdot m} + -1\right) \cdot m \]

   15. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right)} \cdot m \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1 - m}{v}}, m, -1\right) \cdot m \]

   17. lower--.f64 99.7

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 - m}}{v}, m, -1\right) \cdot m \]

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right)} \cdot m \]
6. Add Preprocessing

Alternative 9: 27.0% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ -m \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (- m))

C

double code(double m, double v) {
	return -m;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = -m
end function

Java

public static double code(double m, double v) {
	return -m;
}

Python

def code(m, v):
	return -m

Julia

function code(m, v)
	return Float64(-m)
end

MATLAB

function tmp = code(m, v)
	tmp = -m;
end

Wolfram

code[m_, v_] := (-m)

Derivation

1. Initial program 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
2. Add Preprocessing

3. Taylor expanded in m around 0

   \[\leadsto \color{blue}{-1 \cdot m} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(m\right)} \]

   2. lower-neg.f64 25.0

      \[\leadsto \color{blue}{-m} \]

5. Applied rewrites 25.0%

   \[\leadsto \color{blue}{-m} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) m))