a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.3%

Time: 7.6s

Alternatives: 10

Speedup: 0.1×

• 40.7% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.5e-35) (- (/ m (/ v m)) m) (* (- 1.0 m) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.5e-35) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (1.0 - m) * (m * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 4.5d-35) then
        tmp = (m / (v / m)) - m
    else
        tmp = (1.0d0 - m) * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.5e-35) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (1.0 - m) * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.5e-35:
		tmp = (m / (v / m)) - m
	else:
		tmp = (1.0 - m) * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.5e-35)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.5e-35)
		tmp = (m / (v / m)) - m;
	else
		tmp = (1.0 - m) * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.5e-35], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.5000000000000001e-35

   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%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 91.9%

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

      1. neg-mul-1 91.9%

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

      2. +-commutative 91.9%

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

      3. unsub-neg 91.9%

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

      4. unpow2 91.9%

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

      5. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

   if 4.5000000000000001e-35 < 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%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

      5. neg-mul-1 99.9%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 35.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{3}}{v} + \frac{{m}^{2}}{v}} \]
   7. Step-by-step derivation

      1. mul-1-neg 35.5%

         \[\leadsto \color{blue}{\left(-\frac{{m}^{3}}{v}\right)} + \frac{{m}^{2}}{v} \]

      2. +-commutative 35.5%

         \[\leadsto \color{blue}{\frac{{m}^{2}}{v} + \left(-\frac{{m}^{3}}{v}\right)} \]

      3. unpow2 35.5%

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

      4. associate-*r/ 35.5%

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

      5. unsub-neg 35.5%

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

      6. associate-*r/ 35.5%

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

      7. *-lft-identity 35.5%

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

      8. associate-/l* 35.5%

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

      9. cube-mult 35.5%

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

      10. associate-/l* 35.5%

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

      11. div-sub 99.8%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. associate-/l* 99.9%

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

      15. associate-/r/ 99.8%

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

      16. associate-*r/ 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. distribute-rgt-in 99.8%

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

   2. associate-/r/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. *-commutative 99.8%

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

   5. neg-mul-1 99.8%

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

   6. fma-def 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.1e-17) (- (/ m (/ v m)) m) (* m (/ (- 1.0 m) (/ v m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-17) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * ((1.0 - 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 <= 2.1d-17) then
        tmp = (m / (v / m)) - m
    else
        tmp = m * ((1.0d0 - m) / (v / m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-17) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * ((1.0 - m) / (v / m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.1e-17:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * ((1.0 - m) / (v / m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.1e-17)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m * Float64(Float64(1.0 - m) / Float64(v / m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.1e-17)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * ((1.0 - m) / (v / m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.1e-17], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(N[(1.0 - m), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.09999999999999992e-17

   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%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 92.3%

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

      1. neg-mul-1 92.3%

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

      2. +-commutative 92.3%

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

      3. unsub-neg 92.3%

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

      4. unpow2 92.3%

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

      5. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

   if 2.09999999999999992e-17 < 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* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   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/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r/ 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r/ 99.8%

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

      7. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \]

Alternative 4: 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. 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%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 5: 73.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.25 \cdot 10^{-182}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.25e-182) (- m) (if (<= m 1.0) (* m (/ m v)) (* m (/ (- m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.25e-182) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = m * (-m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.25d-182) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m * (m / v)
    else
        tmp = m * (-m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.25e-182) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = m * (-m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.25e-182:
		tmp = -m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = m * (-m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.25e-182)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(m * Float64(Float64(-m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.25e-182)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = m * (-m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.25e-182], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[((-m) / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.2499999999999999e-182

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 90.2%

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

      1. neg-mul-1 90.2%

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

   6. Simplified 90.2%

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

   if 2.2499999999999999e-182 < 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%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in v around 0 71.6%

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

      1. associate-/l* 71.6%

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

      2. unpow2 71.6%

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

   6. Simplified 71.6%

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

   7. Taylor expanded in m around 0 67.8%

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

      1. associate-*l/ 72.8%

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

   9. Applied egg-rr 72.8%

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

   if 1 < 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* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 0.1%

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

      1. neg-mul-1 0.1%

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

      2. +-commutative 0.1%

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

      3. unsub-neg 0.1%

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

      4. unpow2 0.1%

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

      5. associate-/l* 0.1%

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

   6. Simplified 0.1%

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

      1. sub-neg 0.1%

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

      2. associate-/l* 0.1%

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

      3. pow2 0.1%

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

      4. +-commutative 0.1%

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

      5. pow2 0.1%

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

      6. associate-/l* 0.1%

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

      7. frac-2neg 0.1%

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

      8. distribute-frac-neg 0.1%

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

      9. add-sqr-sqrt 0.1%

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

      10. sqrt-prod 0.1%

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

      11. sqr-neg 0.1%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 74.0%

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

      14. frac-2neg 74.0%

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

      15. associate-/l* 74.0%

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

      16. pow2 74.0%

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

      17. sub-neg 74.0%

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

      18. *-un-lft-identity 74.0%

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

      19. cancel-sign-sub-inv 74.0%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 30.6%

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

      22. sqr-neg 30.6%

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

      23. sqrt-prod 74.0%

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

      24. add-sqr-sqrt 74.0%

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

   8. Applied egg-rr 74.0%

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

      1. neg-mul-1 74.0%

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

      2. sub-neg 74.0%

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

   10. Simplified 74.0%

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

   11. Taylor expanded in m around inf 74.0%

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

       1. mul-1-neg 74.0%

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

       2. unpow2 74.0%

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

       3. associate-*r/ 74.0%

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

   13. Simplified 74.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.25 \cdot 10^{-182}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \]

Alternative 6: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = (m * m) / (-v / 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(Float64(m * m) / Float64(Float64(-v) / 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 * m) / (-v / 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], N[(N[(m * m), $MachinePrecision] / N[((-v) / m), $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. Taylor expanded in m around 0 97.7%

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

   if 1 < 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* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   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%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 95.8%

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

      1. associate-*r/ 95.8%

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

      2. neg-mul-1 95.8%

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

   9. Simplified 95.8%

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

4. Final simplification 96.7%

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

Alternative 7: 38.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-183}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6e-183) (- m) (if (<= m 1.0) (* m (/ m v)) (- m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6e-183) {
		tmp = -m;
	} else if (m <= 1.0) {
		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 (m <= 6d-183) then
        tmp = -m
    else if (m <= 1.0d0) 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 (m <= 6e-183) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6e-183:
		tmp = -m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6e-183)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		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 (m <= 6e-183)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6e-183], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]]

Derivation

1. Split input into 2 regimes

2. if m < 5.9999999999999996e-183 or 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 31.1%

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

      1. neg-mul-1 31.1%

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

   6. Simplified 31.1%

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

   if 5.9999999999999996e-183 < 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%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in v around 0 71.6%

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

      1. associate-/l* 71.6%

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

      2. unpow2 71.6%

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

   6. Simplified 71.6%

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

   7. Taylor expanded in m around 0 67.8%

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

      1. associate-*l/ 72.8%

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

   9. Applied egg-rr 72.8%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-183}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 8: 87.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = m * (-m / v)
	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 * Float64(Float64(-m) / v));
	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 * (-m / v);
	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], N[(m * N[((-m) / v), $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. Taylor expanded in m around 0 97.7%

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

   if 1 < 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* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 0.1%

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

      1. neg-mul-1 0.1%

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

      2. +-commutative 0.1%

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

      3. unsub-neg 0.1%

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

      4. unpow2 0.1%

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

      5. associate-/l* 0.1%

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

   6. Simplified 0.1%

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

      1. sub-neg 0.1%

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

      2. associate-/l* 0.1%

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

      3. pow2 0.1%

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

      4. +-commutative 0.1%

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

      5. pow2 0.1%

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

      6. associate-/l* 0.1%

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

      7. frac-2neg 0.1%

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

      8. distribute-frac-neg 0.1%

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

      9. add-sqr-sqrt 0.1%

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

      10. sqrt-prod 0.1%

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

      11. sqr-neg 0.1%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 74.0%

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

      14. frac-2neg 74.0%

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

      15. associate-/l* 74.0%

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

      16. pow2 74.0%

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

      17. sub-neg 74.0%

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

      18. *-un-lft-identity 74.0%

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

      19. cancel-sign-sub-inv 74.0%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 30.6%

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

      22. sqr-neg 30.6%

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

      23. sqrt-prod 74.0%

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

      24. add-sqr-sqrt 74.0%

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

   8. Applied egg-rr 74.0%

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

      1. neg-mul-1 74.0%

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

      2. sub-neg 74.0%

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

   10. Simplified 74.0%

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

   11. Taylor expanded in m around inf 74.0%

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

       1. mul-1-neg 74.0%

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

       2. unpow2 74.0%

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

       3. associate-*r/ 74.0%

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

   13. Simplified 74.0%

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

4. Final simplification 85.8%

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

Alternative 9: 28.1% 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%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in m around 0 29.1%

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

   1. neg-mul-1 29.1%

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

6. Simplified 29.1%

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

7. Final simplification 29.1%

   \[\leadsto -m \]

Alternative 10: 3.1% 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 99.8%

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

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in m around 0 45.0%

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

   1. neg-mul-1 45.0%

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

   2. +-commutative 45.0%

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

   3. unsub-neg 45.0%

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

   4. unpow2 45.0%

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

   5. associate-/l* 48.5%

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

6. Simplified 48.5%

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

   1. sub-neg 48.5%

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

   2. associate-/l* 45.0%

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

   3. pow2 45.0%

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

   4. +-commutative 45.0%

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

   5. pow2 45.0%

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

   6. associate-/l* 48.5%

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

   7. frac-2neg 48.5%

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

   8. distribute-frac-neg 48.5%

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

   9. add-sqr-sqrt 48.5%

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

   10. sqrt-prod 45.0%

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

   11. sqr-neg 45.0%

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

   12. sqrt-unprod 0.0%

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

   13. add-sqr-sqrt 63.5%

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

   14. frac-2neg 63.5%

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

   15. associate-/l* 63.6%

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

   16. pow2 63.6%

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

   17. sub-neg 63.6%

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

   18. *-un-lft-identity 63.6%

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

   19. cancel-sign-sub-inv 63.6%

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

   20. add-sqr-sqrt 0.0%

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

   21. sqrt-unprod 17.5%

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

   22. sqr-neg 17.5%

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

   23. sqrt-prod 39.2%

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

   24. add-sqr-sqrt 39.2%

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

8. Applied egg-rr 39.1%

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

   1. neg-mul-1 39.1%

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

   2. sub-neg 39.1%

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

10. Simplified 39.1%

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

11. Taylor expanded in m around 0 3.0%

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

12. Final simplification 3.0%

    \[\leadsto m \]

Reproduce

herbie shell --seed 2023285 
(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))