a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.1%

Time: 7.6s

Alternatives: 11

Speedup: 1.0×

• 42.4% 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: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.55 \cdot 10^{-52}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.55e-52) (* m (+ -1.0 (/ m v))) (/ m (/ v (* m (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.55e-52) {
		tmp = m * (-1.0 + (m / v));
	} 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 <= 2.55d-52) then
        tmp = m * ((-1.0d0) + (m / v))
    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 <= 2.55e-52) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m / (v / (m * (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.55e-52:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m / (v / (m * (1.0 - m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.55e-52)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m / Float64(v / Float64(m * Float64(1.0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.55e-52)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = m / (v / (m * (1.0 - m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.55e-52], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.54999999999999995e-52

   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 99.8%

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

   if 2.54999999999999995e-52 < m

   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 v around 0 99.8%

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

      1. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. associate-*r* 99.8%

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

      3. clear-num 99.8%

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

      4. div-inv 99.8%

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

      5. clear-num 99.9%

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

      6. un-div-inv 99.9%

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

      7. associate-/l/ 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.55 \cdot 10^{-52}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.55 \cdot 10^{-52}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.55e-52) (* m (+ -1.0 (/ m v))) (* (- 1.0 m) (/ m (/ v m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.55e-52) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) * (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 <= 2.55d-52) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = (1.0d0 - m) * (m / (v / m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.55e-52) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) * (m / (v / m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.55e-52:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = (1.0 - m) * (m / (v / m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.55e-52)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m / Float64(v / m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.55e-52)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = (1.0 - m) * (m / (v / m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.55e-52], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.54999999999999995e-52

   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 99.8%

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

   if 2.54999999999999995e-52 < m

   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 v around 0 99.8%

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

      1. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. add-sqr-sqrt 14.0%

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

      2. pow2 14.0%

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

      3. sqrt-prod 13.9%

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

      4. sqrt-pow1 13.9%

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

      5. metadata-eval 13.9%

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

      6. pow1 13.9%

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

   9. Applied egg-rr 13.9%

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

       1. unpow2 13.9%

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

       2. swap-sqr 13.9%

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

       3. add-sqr-sqrt 99.8%

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

       4. associate-*r* 99.8%

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

       5. associate-/l* 99.8%

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

       6. clear-num 99.9%

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

       7. *-un-lft-identity 99.9%

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

       8. *-commutative 99.9%

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

       9. times-frac 99.9%

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

       10. clear-num 99.8%

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

       11. div-inv 99.8%

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

       12. clear-num 99.8%

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

       13. associate-/r/ 99.8%

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

       14. div-inv 99.8%

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

       15. metadata-eval 99.8%

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

       16. *-rgt-identity 99.8%

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

       17. associate-*r* 99.8%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.55 \cdot 10^{-52}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.7e-14) (- (/ m (/ v m)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.7e-14) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (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 <= 1.7d-14) then
        tmp = (m / (v / m)) - m
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.7e-14) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.7e-14:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.7e-14)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.7e-14)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.7e-14], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.70000000000000001e-14

   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 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. distribute-rgt-in 99.8%

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

      5. add-sqr-sqrt 99.6%

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

      6. sqrt-unprod 50.5%

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

      7. sqr-neg 50.5%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 46.6%

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

      10. associate-/r/ 46.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 50.6%

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

      13. sqr-neg 50.6%

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

      14. sqrt-unprod 99.5%

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

      15. add-sqr-sqrt 99.8%

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

      16. neg-mul-1 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 1.70000000000000001e-14 < m

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

      5. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-*r/ 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 98.1%

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

      1. *-commutative 98.1%

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

      2. sub-neg 98.1%

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

      3. metadata-eval 98.1%

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

      4. distribute-rgt-in 98.1%

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

      5. add-sqr-sqrt 98.0%

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

      6. sqrt-unprod 48.5%

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

      7. sqr-neg 48.5%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 44.5%

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

      10. associate-/r/ 44.5%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 48.5%

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

      13. sqr-neg 48.5%

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

      14. sqrt-unprod 97.8%

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

      15. add-sqr-sqrt 98.1%

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

      16. neg-mul-1 98.1%

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

   5. Applied egg-rr 98.1%

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

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.9%

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

      5. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in m around inf 99.0%

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

      1. neg-mul-1 99.0%

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

      2. distribute-neg-frac2 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 98.6%

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

Alternative 5: 97.8% accurate, 0.8× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 98.1%

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

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.9%

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

      5. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in m around inf 99.0%

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

      1. neg-mul-1 99.0%

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

      2. distribute-neg-frac2 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 98.6%

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

Alternative 6: 51.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 98.1%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. 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 5.7%

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

      1. neg-mul-1 5.7%

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

   7. Simplified 5.7%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]
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 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(m * N[(N[(m / N[(v / N[(1.0 - m), $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. sub-neg 99.8%

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

   2. associate-*r/ 99.8%

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

   3. clear-num 99.8%

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

   4. un-div-inv 99.8%

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

   5. metadata-eval 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 99.8%

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

Alternative 10: 37.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.35 \cdot 10^{-186}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 1.35e-186) (* m (/ m v)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 1.35e-186) {
		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 <= 1.35d-186) 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 <= 1.35e-186) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 1.35e-186:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 1.35e-186)
		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 <= 1.35e-186)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 1.35e-186], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 1.3499999999999999e-186

   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. sub-neg 99.8%

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

      2. associate-*r/ 99.8%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.8%

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

      5. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 96.1%

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

   6. Taylor expanded in m around 0 46.2%

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

   if 1.3499999999999999e-186 < 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* 99.8%

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

      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 38.8%

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

      1. neg-mul-1 38.8%

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

   7. Simplified 38.8%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.35 \cdot 10^{-186}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.4% 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 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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 25.2%

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

   1. neg-mul-1 25.2%

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

7. Simplified 25.2%

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

Reproduce

herbie shell --seed 2024107 
(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))