a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 10.2s

Alternatives: 10

Speedup: 1.0×

• 42.2% 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: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. distribute-rgt-in 99.9%

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

   2. fma-define 99.9%

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

   3. associate-*r/ 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-/l* 99.9%

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

   6. neg-mul-1 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \frac{m}{v}\\ \mathbf{if}\;m \leq 7.2 \cdot 10^{-147}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.42 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 0.013:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (/ m v))))
   (if (<= m 7.2e-147)
     (- m)
     (if (<= m 1.42e-123)
       t_0
       (if (<= m 4.8e-115) (- m) (if (<= m 0.013) t_0 (* (/ m v) (- m))))))))

C

double code(double m, double v) {
	double t_0 = m * (m / v);
	double tmp;
	if (m <= 7.2e-147) {
		tmp = -m;
	} else if (m <= 1.42e-123) {
		tmp = t_0;
	} else if (m <= 4.8e-115) {
		tmp = -m;
	} else if (m <= 0.013) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = m * (m / v)
    if (m <= 7.2d-147) then
        tmp = -m
    else if (m <= 1.42d-123) then
        tmp = t_0
    else if (m <= 4.8d-115) then
        tmp = -m
    else if (m <= 0.013d0) then
        tmp = t_0
    else
        tmp = (m / v) * -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = m * (m / v);
	double tmp;
	if (m <= 7.2e-147) {
		tmp = -m;
	} else if (m <= 1.42e-123) {
		tmp = t_0;
	} else if (m <= 4.8e-115) {
		tmp = -m;
	} else if (m <= 0.013) {
		tmp = t_0;
	} else {
		tmp = (m / v) * -m;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (m / v)
	tmp = 0
	if m <= 7.2e-147:
		tmp = -m
	elif m <= 1.42e-123:
		tmp = t_0
	elif m <= 4.8e-115:
		tmp = -m
	elif m <= 0.013:
		tmp = t_0
	else:
		tmp = (m / v) * -m
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(m / v))
	tmp = 0.0
	if (m <= 7.2e-147)
		tmp = Float64(-m);
	elseif (m <= 1.42e-123)
		tmp = t_0;
	elseif (m <= 4.8e-115)
		tmp = Float64(-m);
	elseif (m <= 0.013)
		tmp = t_0;
	else
		tmp = Float64(Float64(m / v) * Float64(-m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (m / v);
	tmp = 0.0;
	if (m <= 7.2e-147)
		tmp = -m;
	elseif (m <= 1.42e-123)
		tmp = t_0;
	elseif (m <= 4.8e-115)
		tmp = -m;
	elseif (m <= 0.013)
		tmp = t_0;
	else
		tmp = (m / v) * -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 7.2e-147], (-m), If[LessEqual[m, 1.42e-123], t$95$0, If[LessEqual[m, 4.8e-115], (-m), If[LessEqual[m, 0.013], t$95$0, N[(N[(m / v), $MachinePrecision] * (-m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if m < 7.20000000000000023e-147 or 1.42000000000000008e-123 < m < 4.80000000000000042e-115

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 77.1%

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

      1. neg-mul-1 77.1%

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

   7. Simplified 77.1%

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

   if 7.20000000000000023e-147 < m < 1.42000000000000008e-123 or 4.80000000000000042e-115 < m < 0.0129999999999999994

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in m around 0 95.5%

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

   6. Taylor expanded in m around inf 69.4%

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

      1. unpow2 73.5%

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

      2. associate-/l* 73.5%

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

      3. clear-num 73.4%

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

      4. metadata-eval 73.4%

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

      5. associate-*l/ 73.4%

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

      6. expm1-log1p-u 68.0%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{\frac{v}{m}} \cdot -1\right)\right)}\right) \]

      7. associate-*l/ 68.0%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{-1 \cdot -1}{\frac{v}{m}}}\right)\right)\right) \]

      8. metadata-eval 68.0%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{v}{m}}\right)\right)\right) \]

      9. clear-num 68.0%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{m}{v}}\right)\right)\right) \]

      10. log1p-define 68.2%

          \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{m}{v}\right)}\right)\right) \]

      11. expm1-define 68.2%

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

      12. add-exp-log 73.7%

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

      13. *-commutative 73.7%

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

      14. add-exp-log 68.2%

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

      15. expm1-define 68.2%

          \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{m}{v}\right)\right)} \cdot m\right) \]

      16. log1p-define 68.0%

          \[\leadsto \left(1 - m\right) \cdot \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{m}{v}\right)}\right) \cdot m\right) \]

      17. expm1-log1p-u 73.5%

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

   8. Applied egg-rr 69.4%

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

   if 0.0129999999999999994 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 0.3%

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

   6. Taylor expanded in m around inf 0.3%

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

      1. unpow2 0.3%

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

      2. associate-/l* 0.3%

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

      3. clear-num 0.3%

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

      4. metadata-eval 0.3%

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

      5. associate-*l/ 0.3%

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

      6. associate-*l* 0.3%

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

      7. *-commutative 0.3%

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

      8. associate-*l* 0.3%

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

      9. associate-/r/ 0.3%

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

      10. associate-*l/ 0.3%

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

      11. neg-mul-1 0.3%

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

      12. add-sqr-sqrt 0.3%

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

      13. sqrt-unprod 0.3%

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

      14. sqr-neg 0.3%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 77.3%

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

      17. frac-2neg 77.3%

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

      18. *-commutative 77.3%

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

      19. neg-mul-1 77.3%

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

   8. Applied egg-rr 77.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-147}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.42 \cdot 10^{-123}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 0.013:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 36.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 6.2 \cdot 10^{-200} \lor \neg \left(v \leq 2.3 \cdot 10^{-172}\right) \land v \leq 6 \cdot 10^{-165}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= v 6.2e-200) (and (not (<= v 2.3e-172)) (<= v 6e-165)))
   (* m (/ m v))
   (- m)))

C

double code(double m, double v) {
	double tmp;
	if ((v <= 6.2e-200) || (!(v <= 2.3e-172) && (v <= 6e-165))) {
		tmp = m * (m / v);
	} 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 ((v <= 6.2d-200) .or. (.not. (v <= 2.3d-172)) .and. (v <= 6d-165)) then
        tmp = m * (m / v)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((v <= 6.2e-200) || (!(v <= 2.3e-172) && (v <= 6e-165))) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (v <= 6.2e-200) or (not (v <= 2.3e-172) and (v <= 6e-165)):
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((v <= 6.2e-200) || (!(v <= 2.3e-172) && (v <= 6e-165)))
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((v <= 6.2e-200) || (~((v <= 2.3e-172)) && (v <= 6e-165)))
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[v, 6.2e-200], And[N[Not[LessEqual[v, 2.3e-172]], $MachinePrecision], LessEqual[v, 6e-165]]], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 6.1999999999999998e-200 or 2.29999999999999995e-172 < v < 5.99999999999999958e-165

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 58.5%

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

   6. Taylor expanded in m around inf 32.8%

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

      1. unpow2 74.1%

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

      2. associate-/l* 89.2%

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

      3. clear-num 89.2%

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

      4. metadata-eval 89.2%

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

      5. associate-*l/ 89.2%

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

      6. expm1-log1p-u 85.1%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{\frac{v}{m}} \cdot -1\right)\right)}\right) \]

      7. associate-*l/ 85.1%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{-1 \cdot -1}{\frac{v}{m}}}\right)\right)\right) \]

      8. metadata-eval 85.1%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{v}{m}}\right)\right)\right) \]

      9. clear-num 85.1%

         \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{m}{v}}\right)\right)\right) \]

      10. log1p-define 85.1%

          \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{m}{v}\right)}\right)\right) \]

      11. expm1-define 85.1%

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

      12. add-exp-log 89.2%

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

      13. *-commutative 89.2%

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

      14. add-exp-log 85.1%

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

      15. expm1-define 85.1%

          \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{m}{v}\right)\right)} \cdot m\right) \]

      16. log1p-define 85.1%

          \[\leadsto \left(1 - m\right) \cdot \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{m}{v}\right)}\right) \cdot m\right) \]

      17. expm1-log1p-u 89.2%

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

   8. Applied egg-rr 48.0%

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

   if 6.1999999999999998e-200 < v < 2.29999999999999995e-172 or 5.99999999999999958e-165 < v

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 40.6%

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

      1. neg-mul-1 40.6%

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

   7. Simplified 40.6%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 6.2 \cdot 10^{-200} \lor \neg \left(v \leq 2.3 \cdot 10^{-172}\right) \land v \leq 6 \cdot 10^{-165}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.1 \cdot 10^{-25}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(1 - m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.1e-25) (* m (+ (/ m v) -1.0)) (* (/ m v) (* m (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.1e-25) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = (m / v) * (m * (1.0 - 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 <= 3.1d-25) then
        tmp = m * ((m / v) + (-1.0d0))
    else
        tmp = (m / v) * (m * (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.1e-25) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = (m / v) * (m * (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.1e-25:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = (m / v) * (m * (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.1e-25)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m / v) * Float64(m * Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.1e-25)
		tmp = m * ((m / v) + -1.0);
	else
		tmp = (m / v) * (m * (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.1e-25], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.09999999999999995e-25

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.8%

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

   if 3.09999999999999995e-25 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. fma-define 99.9%

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

      3. associate-*r/ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. neg-mul-1 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.6%

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

   9. Simplified 99.6%

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

       1. unpow2 99.6%

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

       2. associate-*l/ 99.6%

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

       3. associate-/r/ 99.6%

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

       4. associate-*r/ 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. clear-num 99.6%

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

       2. associate-/r/ 99.6%

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

       3. clear-num 99.7%

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

       4. *-commutative 99.7%

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

   13. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.1 \cdot 10^{-25}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(1 - m\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.1 \cdot 10^{-25}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.1e-25) (* m (+ (/ m v) -1.0)) (* (- 1.0 m) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.1e-25) {
		tmp = m * ((m / v) + -1.0);
	} 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 <= 3.1d-25) then
        tmp = m * ((m / v) + (-1.0d0))
    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 <= 3.1e-25) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = (1.0 - m) * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.1e-25:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = (1.0 - m) * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.1e-25)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	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 <= 3.1e-25)
		tmp = m * ((m / v) + -1.0);
	else
		tmp = (1.0 - m) * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.1e-25], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $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 < 3.09999999999999995e-25

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.8%

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

   if 3.09999999999999995e-25 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. fma-define 99.9%

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

      3. associate-*r/ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. neg-mul-1 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.6%

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

   9. Simplified 99.6%

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

       1. unpow2 99.6%

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

       2. associate-/l* 99.6%

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

       3. clear-num 99.6%

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

       4. metadata-eval 99.6%

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

       5. associate-*l/ 99.6%

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

       6. expm1-log1p-u 97.9%

          \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{\frac{v}{m}} \cdot -1\right)\right)}\right) \]

       7. associate-*l/ 97.9%

          \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{-1 \cdot -1}{\frac{v}{m}}}\right)\right)\right) \]

       8. metadata-eval 97.9%

          \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{v}{m}}\right)\right)\right) \]

       9. clear-num 97.9%

          \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{m}{v}}\right)\right)\right) \]

       10. log1p-define 97.9%

           \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{m}{v}\right)}\right)\right) \]

       11. expm1-define 97.9%

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

       12. add-exp-log 99.6%

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

       13. *-commutative 99.6%

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

       14. add-exp-log 97.9%

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

       15. expm1-define 97.9%

           \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{m}{v}\right)\right)} \cdot m\right) \]

       16. log1p-define 97.9%

           \[\leadsto \left(1 - m\right) \cdot \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{m}{v}\right)}\right) \cdot m\right) \]

       17. expm1-log1p-u 99.6%

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

   11. Applied egg-rr 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.1 \cdot 10^{-25}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.00135:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.00135) (* m (+ (/ m v) -1.0)) (* (/ m v) (- m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.00135) {
		tmp = m * ((m / v) + -1.0);
	} 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 <= 0.00135d0) then
        tmp = m * ((m / v) + (-1.0d0))
    else
        tmp = (m / v) * -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.00135) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = (m / v) * -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.00135:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = (m / v) * -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.00135)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m / v) * Float64(-m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.00135)
		tmp = m * ((m / v) + -1.0);
	else
		tmp = (m / v) * -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.00135], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * (-m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.0013500000000000001

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 98.6%

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

   if 0.0013500000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 0.6%

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

   6. Taylor expanded in m around inf 0.6%

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

      1. unpow2 0.6%

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

      2. associate-/l* 0.6%

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

      3. clear-num 0.6%

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

      4. metadata-eval 0.6%

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

      5. associate-*l/ 0.6%

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

      6. associate-*l* 0.6%

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

      7. *-commutative 0.6%

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

      8. associate-*l* 0.6%

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

      9. associate-/r/ 0.6%

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

      10. associate-*l/ 0.6%

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

      11. neg-mul-1 0.6%

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

      12. add-sqr-sqrt 0.6%

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

      13. sqrt-unprod 0.4%

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

      14. sqr-neg 0.4%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 76.7%

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

      17. frac-2neg 76.7%

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

      18. *-commutative 76.7%

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

      19. neg-mul-1 76.7%

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

   8. Applied egg-rr 76.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.00135:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

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(m * Float64(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[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

Alternative 9: 27.0% accurate, 5.5× 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 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 31.1%

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

   1. neg-mul-1 31.1%

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

7. Simplified 31.1%

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

Alternative 10: 3.0% accurate, 11.0× 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 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 51.9%

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

   1. div-inv 51.9%

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

   2. add-sqr-sqrt 51.8%

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

   3. sqrt-unprod 45.6%

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

   4. sqr-neg 45.6%

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

   5. sqrt-unprod 0.0%

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

   6. add-sqr-sqrt 64.8%

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

   7. neg-mul-1 64.8%

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

   8. *-commutative 64.8%

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

   9. associate-*l/ 64.8%

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

   10. clear-num 64.8%

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

   11. associate-*l/ 64.8%

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

   12. metadata-eval 64.8%

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

7. Applied egg-rr 64.8%

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

8. Taylor expanded in v around 0 52.7%

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

   1. distribute-lft-out 52.7%

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

   2. mul-1-neg 52.7%

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

   3. unpow2 52.7%

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

   4. distribute-lft-out 52.7%

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

   5. distribute-rgt-neg-in 52.7%

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

10. Simplified 52.7%

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

    1. add-sqr-sqrt 0.0%

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

    2. sqrt-unprod 17.8%

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

    3. sqr-neg 17.8%

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

    4. sqrt-unprod 17.8%

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

    5. add-sqr-sqrt 17.8%

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

    6. +-commutative 17.8%

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

    7. distribute-rgt-in 17.8%

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

    8. pow2 17.8%

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

12. Applied egg-rr 17.8%

    \[\leadsto \frac{\color{blue}{{m}^{2} + v \cdot m}}{v} \]
13. Step-by-step derivation

    1. unpow2 17.8%

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

    2. distribute-rgt-in 17.8%

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

14. Simplified 17.8%

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

15. Taylor expanded in m around 0 3.1%

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

Reproduce

herbie shell --seed 2024105 
(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))