a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.7%

Time: 6.2s

Alternatives: 8

Speedup: 1.0×

• 41.4% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.4 \cdot 10^{-21}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{m}}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.4e-21) (- (* m (/ m v)) m) (/ m (/ (/ v m) (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.4e-21) {
		tmp = (m * (m / v)) - m;
	} 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 <= 3.4d-21) then
        tmp = (m * (m / v)) - m
    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 <= 3.4e-21) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m / ((v / m) / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.4e-21:
		tmp = (m * (m / v)) - m
	else:
		tmp = m / ((v / m) / (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.4e-21)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(m / Float64(Float64(v / m) / Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.4e-21)
		tmp = (m * (m / v)) - m;
	else
		tmp = m / ((v / m) / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.4e-21], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m / N[(N[(v / m), $MachinePrecision] / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.4e-21

   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/ 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 82.7%

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

      1. neg-mul-1 82.7%

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

      2. +-commutative 82.7%

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

      3. unsub-neg 82.7%

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

   6. Simplified 82.7%

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

      1. div-inv 82.7%

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

      2. unpow2 82.7%

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

      3. associate-*l* 99.0%

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

      4. div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 3.4e-21 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. unpow2 99.9%

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

      3. clear-num 99.9%

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

      4. associate-*r* 99.9%

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

      5. add-sqr-sqrt 5.5%

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

      6. associate-*r* 5.4%

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

      7. *-commutative 5.4%

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

      8. associate-/r/ 5.4%

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

      9. div-inv 5.4%

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

      10. clear-num 5.4%

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

      11. *-commutative 5.4%

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

      12. associate-/r/ 5.4%

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

      13. div-inv 5.4%

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

      14. clear-num 5.4%

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

   8. Applied egg-rr 5.4%

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

      1. associate-*l* 5.5%

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

      2. add-sqr-sqrt 99.9%

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

      3. associate-*r/ 99.9%

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

      4. remove-double-div 99.9%

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

      5. associate-/l/ 99.8%

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

      6. associate-/r* 99.9%

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

      7. un-div-inv 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-/l/ 99.9%

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

      10. un-div-inv 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in v around 0 99.9%

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

       1. associate-/r* 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.4 \cdot 10^{-21}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{m}}{1 - m}}\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-22}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-22) (- (* m (/ m v)) m) (* m (* (/ m v) (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-22) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * ((m / v) * (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 <= 5d-22) then
        tmp = (m * (m / v)) - m
    else
        tmp = m * ((m / v) * (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5e-22) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * ((m / v) * (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5e-22:
		tmp = (m * (m / v)) - m
	else:
		tmp = m * ((m / v) * (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-22)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(m * Float64(Float64(m / v) * Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-22)
		tmp = (m * (m / v)) - m;
	else
		tmp = m * ((m / v) * (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-22], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.99999999999999954e-22

   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/ 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 82.7%

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

      1. neg-mul-1 82.7%

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

      2. +-commutative 82.7%

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

      3. unsub-neg 82.7%

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

   6. Simplified 82.7%

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

      1. div-inv 82.7%

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

      2. unpow2 82.7%

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

      3. associate-*l* 99.0%

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

      4. div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 4.99999999999999954e-22 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

      1. associate-/r/ 99.9%

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

      2. *-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-/r/ 99.9%

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

      6. associate-*l* 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-22}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right)\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.05 \cdot 10^{-21}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.05e-21) (- (* m (/ m v)) m) (/ m (/ v (* m (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.05e-21) {
		tmp = (m * (m / v)) - m;
	} 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 <= 1.05d-21) then
        tmp = (m * (m / v)) - m
    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 <= 1.05e-21) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m / (v / (m * (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.05e-21:
		tmp = (m * (m / v)) - m
	else:
		tmp = m / (v / (m * (1.0 - m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.05e-21)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	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 <= 1.05e-21)
		tmp = (m * (m / v)) - m;
	else
		tmp = m / (v / (m * (1.0 - m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.05e-21], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $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 < 1.05000000000000006e-21

   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/ 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 82.7%

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

      1. neg-mul-1 82.7%

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

      2. +-commutative 82.7%

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

      3. unsub-neg 82.7%

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

   6. Simplified 82.7%

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

      1. div-inv 82.7%

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

      2. unpow2 82.7%

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

      3. associate-*l* 99.0%

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

      4. div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 1.05000000000000006e-21 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. unpow2 99.9%

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

      3. clear-num 99.9%

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

      4. associate-*r* 99.9%

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

      5. add-sqr-sqrt 5.5%

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

      6. associate-*r* 5.4%

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

      7. *-commutative 5.4%

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

      8. associate-/r/ 5.4%

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

      9. div-inv 5.4%

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

      10. clear-num 5.4%

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

      11. *-commutative 5.4%

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

      12. associate-/r/ 5.4%

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

      13. div-inv 5.4%

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

      14. clear-num 5.4%

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

   8. Applied egg-rr 5.4%

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

      1. associate-*l* 5.5%

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

      2. add-sqr-sqrt 99.9%

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

      3. associate-*r/ 99.9%

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

      4. remove-double-div 99.9%

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

      5. associate-/l/ 99.8%

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

      6. associate-/r* 99.9%

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

      7. un-div-inv 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-/l/ 99.9%

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

      10. un-div-inv 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.05 \cdot 10^{-21}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

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

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 5: 52.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. 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. associate-*r/ 99.0%

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

      3. fma-neg 99.0%

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

      4. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in m around 0 98.5%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 5.8%

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

      1. neg-mul-1 5.8%

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

   6. Simplified 5.8%

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

4. Final simplification 55.0%

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

Alternative 6: 52.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. 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/ 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 82.4%

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

      1. neg-mul-1 82.4%

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

      2. +-commutative 82.4%

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

      3. unsub-neg 82.4%

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

   6. Simplified 82.4%

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

      1. div-inv 82.3%

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

      2. unpow2 82.3%

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

      3. associate-*l* 97.8%

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

      4. div-inv 98.5%

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

   8. Applied egg-rr 98.5%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 5.8%

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

      1. neg-mul-1 5.8%

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

   6. Simplified 5.8%

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

4. Final simplification 55.1%

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

Alternative 7: 38.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 5.1 \cdot 10^{-172}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 5.1e-172) (* m (/ m v)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 5.1e-172) {
		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 <= 5.1d-172) 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 <= 5.1e-172) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 5.1e-172:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 5.1e-172)
		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 <= 5.1e-172)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 5.1e-172], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 5.0999999999999998e-172

   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/ 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in v around 0 71.4%

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

      1. associate-/l* 71.4%

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

   6. Simplified 71.4%

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

      1. associate-/r/ 71.4%

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

      2. *-commutative 71.4%

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

      3. unpow2 71.4%

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

      4. associate-/l* 87.5%

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

      5. associate-/r/ 87.5%

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

      6. associate-*l* 87.5%

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

   8. Applied egg-rr 87.5%

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

   9. Taylor expanded in m around 0 44.2%

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

   if 5.0999999999999998e-172 < v

   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/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 38.7%

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

      1. neg-mul-1 38.7%

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

   6. Simplified 38.7%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 5.1 \cdot 10^{-172}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 8: 27.3% 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%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in m around 0 27.5%

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

   1. neg-mul-1 27.5%

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

6. Simplified 27.5%

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

7. Final simplification 27.5%

   \[\leadsto -m \]

Reproduce

herbie shell --seed 2023310 
(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))