a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.6%

Time: 55.3s

Alternatives: 12

Speedup: 1.0×

• 40.9% 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: 99.8% 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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{-24}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.8e-24) (* m (+ -1.0 (/ m v))) (/ (* m (- m (* m m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.8e-24) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (m * (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 <= 6.8d-24) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = (m * (m - (m * m))) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 6.8e-24) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (m * (m - (m * m))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.8e-24:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = (m * (m - (m * m))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.8e-24)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(m - Float64(m * m))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.8e-24)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = (m * (m - (m * m))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.8e-24], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.79999999999999985e-24

   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 6.79999999999999985e-24 < 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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

         \[\leadsto {m}^{3} \cdot \left(\color{blue}{\left(0 - \frac{1}{v}\right)} + \frac{1}{m \cdot v}\right) \]

      4. associate-+l- N/A

         \[\leadsto {m}^{3} \cdot \color{blue}{\left(0 - \left(\frac{1}{v} - \frac{1}{m \cdot v}\right)\right)} \]

      5. neg-sub0 N/A

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

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

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. neg-sub0 N/A

          \[\leadsto {m}^{2} \cdot \left(m \cdot \color{blue}{\left(0 - \left(\frac{1}{v} - \frac{1}{m \cdot v}\right)\right)}\right) \]

      13. associate-+l- N/A

          \[\leadsto {m}^{2} \cdot \left(m \cdot \color{blue}{\left(\left(0 - \frac{1}{v}\right) + \frac{1}{m \cdot v}\right)}\right) \]

      14. neg-sub0 N/A

          \[\leadsto {m}^{2} \cdot \left(m \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{1}{m \cdot v}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto {m}^{2} \cdot \left(m \cdot \color{blue}{\left(\frac{1}{m \cdot v} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)}\right) \]

      16. distribute-lft-in N/A

          \[\leadsto {m}^{2} \cdot \color{blue}{\left(m \cdot \frac{1}{m \cdot v} + m \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)} \]

   5. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{-24}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+26}:\\ \;\;\;\;m \cdot \left(-\frac{m}{v}\right)\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-306}:\\ \;\;\;\;\frac{m \cdot m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (+ (/ (* m (- 1.0 m)) v) -1.0))))
   (if (<= t_0 -2e+26)
     (* m (- (/ m v)))
     (if (<= t_0 -2e-306) (- (/ (* m m) v) m) (* m (/ m v))))))

C

double code(double m, double v) {
	double t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	double tmp;
	if (t_0 <= -2e+26) {
		tmp = m * -(m / v);
	} else if (t_0 <= -2e-306) {
		tmp = ((m * m) / v) - 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) :: t_0
    real(8) :: tmp
    t_0 = m * (((m * (1.0d0 - m)) / v) + (-1.0d0))
    if (t_0 <= (-2d+26)) then
        tmp = m * -(m / v)
    else if (t_0 <= (-2d-306)) then
        tmp = ((m * m) / v) - m
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	double tmp;
	if (t_0 <= -2e+26) {
		tmp = m * -(m / v);
	} else if (t_0 <= -2e-306) {
		tmp = ((m * m) / v) - m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0)
	tmp = 0
	if t_0 <= -2e+26:
		tmp = m * -(m / v)
	elif t_0 <= -2e-306:
		tmp = ((m * m) / v) - m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0))
	tmp = 0.0
	if (t_0 <= -2e+26)
		tmp = Float64(m * Float64(-Float64(m / v)));
	elseif (t_0 <= -2e-306)
		tmp = Float64(Float64(Float64(m * m) / v) - m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	tmp = 0.0;
	if (t_0 <= -2e+26)
		tmp = m * -(m / v);
	elseif (t_0 <= -2e-306)
		tmp = ((m * m) / v) - m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+26], N[(m * (-N[(m / v), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$0, -2e-306], N[(N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -2.0000000000000001e26

   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

      \[\leadsto \color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)} \cdot m \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right)} \cdot m \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{m \cdot \left(m \cdot \frac{1}{m \cdot v}\right)} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      4. associate-/r* N/A

         \[\leadsto \left(m \cdot \left(m \cdot \color{blue}{\frac{\frac{1}{m}}{v}}\right) - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      5. associate-*r/ N/A

         \[\leadsto \left(m \cdot \color{blue}{\frac{m \cdot \frac{1}{m}}{v}} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left(m \cdot \frac{\color{blue}{1}}{v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      7. associate-*r/ N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \color{blue}{\frac{{m}^{2} \cdot 1}{v}}\right) \cdot m \]

      8. *-rgt-identity N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \frac{\color{blue}{{m}^{2}}}{v}\right) \cdot m \]

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

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

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

      16. *-rgt-identity N/A

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

      17. unpow2 N/A

          \[\leadsto \frac{m - \color{blue}{{m}^{2}}}{v} \cdot m \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{m - {m}^{2}}}{v} \cdot m \]

      19. unpow2 N/A

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

      20. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 0.1%

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

   10. Applied rewrites 85.3%

       \[\leadsto \frac{m}{-v} \cdot m \]

   if -2.0000000000000001e26 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -2.00000000000000006e-306

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

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

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{{m}^{2}}}{v} - m \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{{m}^{2}}{v} - \color{blue}{m} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v} - m} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} - m \]

      7. unpow2 N/A

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

      8. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.00000000000000006e-306 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.5%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)} \cdot m \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right)} \cdot m \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{m \cdot \left(m \cdot \frac{1}{m \cdot v}\right)} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      4. associate-/r* N/A

         \[\leadsto \left(m \cdot \left(m \cdot \color{blue}{\frac{\frac{1}{m}}{v}}\right) - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      5. associate-*r/ N/A

         \[\leadsto \left(m \cdot \color{blue}{\frac{m \cdot \frac{1}{m}}{v}} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left(m \cdot \frac{\color{blue}{1}}{v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      7. associate-*r/ N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \color{blue}{\frac{{m}^{2} \cdot 1}{v}}\right) \cdot m \]

      8. *-rgt-identity N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \frac{\color{blue}{{m}^{2}}}{v}\right) \cdot m \]

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

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

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

      16. *-rgt-identity N/A

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

      17. unpow2 N/A

          \[\leadsto \frac{m - \color{blue}{{m}^{2}}}{v} \cdot m \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{m - {m}^{2}}}{v} \cdot m \]

      19. unpow2 N/A

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

      20. lower-*.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 91.8%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+26}:\\ \;\;\;\;m \cdot \left(-\frac{m}{v}\right)\\ \mathbf{elif}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-306}:\\ \;\;\;\;\frac{m \cdot m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+26}:\\ \;\;\;\;m \cdot \left(-\frac{m}{v}\right)\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-306}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (+ (/ (* m (- 1.0 m)) v) -1.0))))
   (if (<= t_0 -2e+26)
     (* m (- (/ m v)))
     (if (<= t_0 -2e-306) (- m) (* m (/ m v))))))

C

double code(double m, double v) {
	double t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	double tmp;
	if (t_0 <= -2e+26) {
		tmp = m * -(m / v);
	} else if (t_0 <= -2e-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) :: t_0
    real(8) :: tmp
    t_0 = m * (((m * (1.0d0 - m)) / v) + (-1.0d0))
    if (t_0 <= (-2d+26)) then
        tmp = m * -(m / v)
    else if (t_0 <= (-2d-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 t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	double tmp;
	if (t_0 <= -2e+26) {
		tmp = m * -(m / v);
	} else if (t_0 <= -2e-306) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0)
	tmp = 0
	if t_0 <= -2e+26:
		tmp = m * -(m / v)
	elif t_0 <= -2e-306:
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0))
	tmp = 0.0
	if (t_0 <= -2e+26)
		tmp = Float64(m * Float64(-Float64(m / v)));
	elseif (t_0 <= -2e-306)
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	tmp = 0.0;
	if (t_0 <= -2e+26)
		tmp = m * -(m / v);
	elseif (t_0 <= -2e-306)
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+26], N[(m * (-N[(m / v), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$0, -2e-306], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -2.0000000000000001e26

   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

      \[\leadsto \color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)} \cdot m \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right)} \cdot m \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{m \cdot \left(m \cdot \frac{1}{m \cdot v}\right)} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      4. associate-/r* N/A

         \[\leadsto \left(m \cdot \left(m \cdot \color{blue}{\frac{\frac{1}{m}}{v}}\right) - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      5. associate-*r/ N/A

         \[\leadsto \left(m \cdot \color{blue}{\frac{m \cdot \frac{1}{m}}{v}} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left(m \cdot \frac{\color{blue}{1}}{v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      7. associate-*r/ N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \color{blue}{\frac{{m}^{2} \cdot 1}{v}}\right) \cdot m \]

      8. *-rgt-identity N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \frac{\color{blue}{{m}^{2}}}{v}\right) \cdot m \]

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

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

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

      16. *-rgt-identity N/A

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

      17. unpow2 N/A

          \[\leadsto \frac{m - \color{blue}{{m}^{2}}}{v} \cdot m \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{m - {m}^{2}}}{v} \cdot m \]

      19. unpow2 N/A

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

      20. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 0.1%

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

   10. Applied rewrites 85.3%

       \[\leadsto \frac{m}{-v} \cdot m \]

   if -2.0000000000000001e26 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -2.00000000000000006e-306

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   if -2.00000000000000006e-306 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.5%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)} \cdot m \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right)} \cdot m \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{m \cdot \left(m \cdot \frac{1}{m \cdot v}\right)} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      4. associate-/r* N/A

         \[\leadsto \left(m \cdot \left(m \cdot \color{blue}{\frac{\frac{1}{m}}{v}}\right) - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      5. associate-*r/ N/A

         \[\leadsto \left(m \cdot \color{blue}{\frac{m \cdot \frac{1}{m}}{v}} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left(m \cdot \frac{\color{blue}{1}}{v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      7. associate-*r/ N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \color{blue}{\frac{{m}^{2} \cdot 1}{v}}\right) \cdot m \]

      8. *-rgt-identity N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \frac{\color{blue}{{m}^{2}}}{v}\right) \cdot m \]

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

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

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

      16. *-rgt-identity N/A

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

      17. unpow2 N/A

          \[\leadsto \frac{m - \color{blue}{{m}^{2}}}{v} \cdot m \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{m - {m}^{2}}}{v} \cdot m \]

      19. unpow2 N/A

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

      20. lower-*.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 91.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+26}:\\ \;\;\;\;m \cdot \left(-\frac{m}{v}\right)\\ \mathbf{elif}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-306}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+26}:\\ \;\;\;\;m \cdot \left(-\frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e+26)
   (* m (- (/ m v)))
   (* m (+ -1.0 (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+26) {
		tmp = m * -(m / v);
	} else {
		tmp = m * (-1.0 + (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 * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-2d+26)) then
        tmp = m * -(m / v)
    else
        tmp = m * ((-1.0d0) + (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+26) {
		tmp = m * -(m / v);
	} else {
		tmp = m * (-1.0 + (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+26:
		tmp = m * -(m / v)
	else:
		tmp = m * (-1.0 + (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e+26)
		tmp = Float64(m * Float64(-Float64(m / v)));
	else
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+26)
		tmp = m * -(m / v);
	else
		tmp = m * (-1.0 + (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e+26], N[(m * (-N[(m / v), $MachinePrecision])), $MachinePrecision], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $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) < -2.0000000000000001e26

   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

      \[\leadsto \color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)} \cdot m \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right)} \cdot m \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{m \cdot \left(m \cdot \frac{1}{m \cdot v}\right)} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      4. associate-/r* N/A

         \[\leadsto \left(m \cdot \left(m \cdot \color{blue}{\frac{\frac{1}{m}}{v}}\right) - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      5. associate-*r/ N/A

         \[\leadsto \left(m \cdot \color{blue}{\frac{m \cdot \frac{1}{m}}{v}} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left(m \cdot \frac{\color{blue}{1}}{v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      7. associate-*r/ N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \color{blue}{\frac{{m}^{2} \cdot 1}{v}}\right) \cdot m \]

      8. *-rgt-identity N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \frac{\color{blue}{{m}^{2}}}{v}\right) \cdot m \]

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

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

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

      16. *-rgt-identity N/A

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

      17. unpow2 N/A

          \[\leadsto \frac{m - \color{blue}{{m}^{2}}}{v} \cdot m \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{m - {m}^{2}}}{v} \cdot m \]

      19. unpow2 N/A

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

      20. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 0.1%

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

   10. Applied rewrites 85.3%

       \[\leadsto \frac{m}{-v} \cdot m \]

   if -2.0000000000000001e26 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   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 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+26}:\\ \;\;\;\;m \cdot \left(-\frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-306}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e-306) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-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 * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-2d-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 * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-306) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-306:
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e-306)
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-306)
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e-306], (-m), N[(m * N[(m / v), $MachinePrecision]), $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) < -2.00000000000000006e-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 34.2

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

   5. Applied rewrites 34.2%

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

   if -2.00000000000000006e-306 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.5%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)} \cdot m \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right)} \cdot m \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{m \cdot \left(m \cdot \frac{1}{m \cdot v}\right)} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      4. associate-/r* N/A

         \[\leadsto \left(m \cdot \left(m \cdot \color{blue}{\frac{\frac{1}{m}}{v}}\right) - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      5. associate-*r/ N/A

         \[\leadsto \left(m \cdot \color{blue}{\frac{m \cdot \frac{1}{m}}{v}} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      6. rgt-mult-inverse N/A

         \[\leadsto \left(m \cdot \frac{\color{blue}{1}}{v} - {m}^{2} \cdot \frac{1}{v}\right) \cdot m \]

      7. associate-*r/ N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \color{blue}{\frac{{m}^{2} \cdot 1}{v}}\right) \cdot m \]

      8. *-rgt-identity N/A

         \[\leadsto \left(m \cdot \frac{1}{v} - \frac{\color{blue}{{m}^{2}}}{v}\right) \cdot m \]

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

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

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

      16. *-rgt-identity N/A

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

      17. unpow2 N/A

          \[\leadsto \frac{m - \color{blue}{{m}^{2}}}{v} \cdot m \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{m - {m}^{2}}}{v} \cdot m \]

      19. unpow2 N/A

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

      20. lower-*.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 91.8%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-306}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. distribute-rgt-in N/A

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

   12. *-lft-identity N/A

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

   13. cancel-sub-sign N/A

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

   14. lower--.f64 N/A

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

   15. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 7: 97.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = m * (-1.0 + (m / v));
	} 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 * ((-1.0d0) + (m / v))
    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 * (-1.0 + (m / v));
	} else {
		tmp = (m * (m * m)) / -v;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m * m), $MachinePrecision]), $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 97.6

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

   5. Applied rewrites 97.6%

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

   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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

         \[\leadsto \frac{m \cdot \color{blue}{{m}^{2}}}{-1 \cdot v} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{m \cdot {m}^{2}}}{-1 \cdot v} \]

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 97.8%

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

Alternative 8: 97.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = m * (-1.0 + (m / v));
	} 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 * ((-1.0d0) + (m / v))
    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 * (-1.0 + (m / v));
	} else {
		tmp = -((m * m) * (m / v));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(m * m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $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 97.6

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

   5. Applied rewrites 97.6%

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

   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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

         \[\leadsto \frac{m \cdot \color{blue}{{m}^{2}}}{-1 \cdot v} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{m \cdot {m}^{2}}}{-1 \cdot v} \]

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.0%

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

4. Final simplification 97.8%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 10: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 11: 26.8% 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 25.5

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

5. Applied rewrites 25.5%

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

Alternative 12: 3.1% accurate, 28.0× 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 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 25.5

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

5. Applied rewrites 25.5%

   \[\leadsto \color{blue}{-m} \]
6. Step-by-step derivation

7. Applied rewrites 34.8%

   \[\leadsto \frac{m \cdot \left(-m\right)}{\color{blue}{0 + m}} \]
8. Step-by-step derivation

9. Applied rewrites 3.2%

   \[\leadsto m \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024278 
(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))