a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 12

Speedup: 1.0×

• 41.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% 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 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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. distribute-lft-in 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-*l/ 93.9%

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

   6. associate-*r/ 99.8%

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

   7. *-lft-identity 99.8%

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

   8. associate-*l/ 99.8%

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

   9. associate-*r* 99.8%

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

   10. *-commutative 99.8%

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

   11. distribute-rgt-out 99.8%

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

   12. associate-*r/ 99.9%

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

   13. associate-/l* 99.9%

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

   14. /-rgt-identity 99.9%

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

   15. associate-/l* 99.9%

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

   16. 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. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{\left(-m\right) - -1}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.8e-15) (- (/ m (/ v m)) m) (* m (/ (- (- m) -1.0) (/ v m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.8e-15) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * ((-m - -1.0) / (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 <= 2.8d-15) then
        tmp = (m / (v / m)) - m
    else
        tmp = m * ((-m - (-1.0d0)) / (v / m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.8e-15) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * ((-m - -1.0) / (v / m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.8e-15:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * ((-m - -1.0) / (v / m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.8e-15)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m * Float64(Float64(Float64(-m) - -1.0) / Float64(v / m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.8e-15)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * ((-m - -1.0) / (v / m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.8e-15], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(N[((-m) - -1.0), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.80000000000000014e-15

   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. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 87.1%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. 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 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-lft-in 99.7%

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

      4. clear-num 99.7%

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

      5. un-div-inv 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 2.80000000000000014e-15 < 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. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      16. 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-*r/ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 100.0%

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

      3. frac-2neg 100.0%

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

      4. frac-2neg 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 3.8%

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

      10. sqr-neg 3.8%

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

      11. sqrt-unprod 3.8%

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

      12. add-sqr-sqrt 3.8%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 100.0%

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

      15. sqr-neg 100.0%

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

      16. sqrt-unprod 99.9%

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

      17. add-sqr-sqrt 100.0%

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

      18. frac-2neg 100.0%

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

      19. distribute-neg-frac 100.0%

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

      20. remove-double-neg 100.0%

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

      21. distribute-frac-neg 100.0%

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

      22. frac-2neg 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{\left(-m\right) - -1}{\frac{v}{m}}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{-15}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.8e-15) (* m (+ -1.0 (/ m v))) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.8e-15) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 6.8d-15) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 6.8e-15) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.8e-15:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.8e-15)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.8e-15)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.8e-15], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.8000000000000001e-15

   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. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 87.1%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. 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 99.8%

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

   if 6.8000000000000001e-15 < 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. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      16. 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-*r/ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{-15}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.22 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.22e-14) (- (/ m (/ v m)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.22e-14) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.22d-14) then
        tmp = (m / (v / m)) - m
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.22e-14) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.22e-14:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.22e-14)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.22e-14)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.22e-14], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.21999999999999994e-14

   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. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 87.1%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. 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 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-lft-in 99.7%

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

      4. clear-num 99.7%

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

      5. un-div-inv 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 1.21999999999999994e-14 < 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. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      16. 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-*r/ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.22 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 5: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.5e-15) (- (/ m (/ v m)) m) (/ (* m m) (/ v (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.5e-15) {
		tmp = (m / (v / m)) - 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 <= 6.5d-15) then
        tmp = (m / (v / m)) - 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 <= 6.5e-15) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (m * m) / (v / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.5e-15:
		tmp = (m / (v / m)) - m
	else:
		tmp = (m * m) / (v / (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.5e-15)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(Float64(m * m) / Float64(v / Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.5e-15)
		tmp = (m / (v / m)) - m;
	else
		tmp = (m * m) / (v / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.5e-15], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m * m), $MachinePrecision] / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.49999999999999991e-15

   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. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 87.1%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. 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 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-lft-in 99.7%

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

      4. clear-num 99.7%

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

      5. un-div-inv 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 6.49999999999999991e-15 < 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. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      16. 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%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. distribute-lft-in 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-*l/ 93.9%

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

   6. associate-*r/ 99.8%

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

   7. *-lft-identity 99.8%

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

   8. associate-*l/ 99.8%

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

   9. associate-*r* 99.8%

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

   10. *-commutative 99.8%

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

   11. distribute-rgt-out 99.8%

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

   12. associate-*r/ 99.9%

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

   13. associate-/l* 99.9%

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

   14. /-rgt-identity 99.9%

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

   15. associate-*l/ 99.9%

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

   16. 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. Final simplification 99.9%

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

Alternative 7: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8 \cdot 10^{-172}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-\frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 8e-172) (- m) (if (<= m 1.0) (/ m (/ v m)) (* m (- (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8e-172) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = m * -(m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 8d-172) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m / (v / m)
    else
        tmp = m * -(m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 8e-172) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = m * -(m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8e-172:
		tmp = -m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = m * -(m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8e-172)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(m * Float64(-Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 8e-172)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = m * -(m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 8e-172], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(m * (-N[(m / v), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 8.0000000000000003e-172

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 83.3%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 83.3%

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

      1. neg-mul-1 83.3%

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

   6. Simplified 83.3%

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

   if 8.0000000000000003e-172 < m < 1

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*l/ 92.2%

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

      6. associate-*r/ 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.5%

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

      9. associate-*r* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out 99.5%

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

      12. associate-*r/ 99.6%

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

      13. associate-/l* 99.6%

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

      14. /-rgt-identity 99.6%

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

      15. associate-*l/ 99.7%

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

      16. 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 63.5%

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

      1. associate-*r/ 63.4%

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

      2. unpow2 63.4%

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

      3. associate-*r* 70.9%

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

   6. Simplified 70.9%

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

      1. *-commutative 70.9%

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

      2. associate-/r/ 71.0%

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

      3. div-inv 70.9%

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

      4. clear-num 71.0%

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

      5. associate-*r* 71.0%

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

      6. clear-num 70.9%

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

      7. un-div-inv 71.1%

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

      8. sub-neg 71.1%

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

      9. distribute-rgt-in 71.0%

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

      10. *-un-lft-identity 71.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 65.3%

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

      13. sqr-neg 65.3%

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

      14. sqrt-unprod 65.3%

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

      15. add-sqr-sqrt 65.3%

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

   8. Applied egg-rr 65.3%

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

   9. Taylor expanded in m around 0 65.5%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-*l/ 99.9%

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

      16. 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-*r/ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. unpow2 0.1%

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

      2. associate-*r/ 0.1%

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

   9. Simplified 0.1%

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

       1. add-sqr-sqrt 0.1%

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

       2. sqrt-prod 0.1%

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

       3. sqr-neg 0.1%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 79.2%

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

       6. distribute-frac-neg 79.2%

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

       7. distribute-rgt-neg-out 79.2%

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

       8. clear-num 79.2%

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

       9. un-div-inv 79.2%

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

   11. Applied egg-rr 79.2%

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

       1. associate-/r/ 79.2%

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

   13. Applied egg-rr 79.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8 \cdot 10^{-172}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-\frac{m}{v}\right)\\ \end{array} \]

Alternative 8: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. 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. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 87.9%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. 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 96.9%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-*l/ 99.9%

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

      16. 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-*r/ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 98.3%

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

      1. neg-mul-1 98.3%

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

      2. distribute-neg-frac 98.3%

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

   9. Simplified 98.3%

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

       1. associate-*r* 98.3%

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

       2. distribute-frac-neg 98.3%

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

       3. distribute-rgt-neg-out 98.3%

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

       4. add-sqr-sqrt 98.2%

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

       5. sqrt-prod 98.3%

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

       6. sqr-neg 98.3%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 0.0%

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

       9. associate-*r* 0.0%

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

       10. add-sqr-sqrt 0.0%

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

       11. sqrt-unprod 92.2%

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

       12. swap-sqr 92.2%

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

       13. distribute-frac-neg 92.2%

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

       14. distribute-frac-neg 92.2%

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

       15. sqr-neg 92.2%

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

       16. swap-sqr 92.2%

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

       17. sqrt-unprod 98.3%

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

       18. add-sqr-sqrt 98.3%

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

       19. clear-num 98.3%

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

       20. un-div-inv 98.3%

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

   11. Applied egg-rr 98.3%

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

4. Final simplification 97.6%

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

Alternative 9: 87.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * (-N[(m / 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. 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. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 87.9%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. 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 96.9%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-*l/ 99.9%

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

      16. 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-*r/ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. unpow2 0.1%

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

      2. associate-*r/ 0.1%

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

   9. Simplified 0.1%

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

       1. add-sqr-sqrt 0.1%

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

       2. sqrt-prod 0.1%

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

       3. sqr-neg 0.1%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 79.2%

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

       6. distribute-frac-neg 79.2%

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

       7. distribute-rgt-neg-out 79.2%

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

       8. clear-num 79.2%

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

       9. un-div-inv 79.2%

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

   11. Applied egg-rr 79.2%

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

       1. associate-/r/ 79.2%

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

   13. Applied egg-rr 79.2%

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

4. Final simplification 88.1%

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

Alternative 10: 37.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 2.9 \cdot 10^{-189}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 2.9e-189) (* m (/ m v)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 2.9e-189) {
		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 <= 2.9d-189) 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 <= 2.9e-189) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 2.9e-189:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 2.9e-189)
		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 <= 2.9e-189)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 2.9e-189], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 2.9e-189

   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. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 85.5%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. 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 v around 0 77.8%

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

      1. associate-*r/ 77.8%

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

      2. unpow2 77.8%

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

      3. associate-*r* 89.5%

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

   6. Simplified 89.5%

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

   7. Taylor expanded in m around 0 27.0%

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

      1. unpow2 27.0%

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

      2. associate-*r/ 38.7%

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

   9. Simplified 38.7%

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

   if 2.9e-189 < v

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      16. 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 42.8%

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

      1. neg-mul-1 42.8%

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

   6. Simplified 42.8%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 2.9 \cdot 10^{-189}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 11: 37.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 4.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 4.2e-188) (/ m (/ v m)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 4.2e-188) {
		tmp = 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 (v <= 4.2d-188) then
        tmp = m / (v / m)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 4.2e-188) {
		tmp = m / (v / m);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 4.2e-188:
		tmp = m / (v / m)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 4.2e-188)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 4.2e-188)
		tmp = m / (v / m);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 4.2e-188], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 4.1999999999999998e-188

   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. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 85.5%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. 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 v around 0 77.8%

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

      1. associate-*r/ 77.8%

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

      2. unpow2 77.8%

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

      3. associate-*r* 89.5%

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

   6. Simplified 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-/r/ 89.6%

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

      3. div-inv 89.5%

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

      4. clear-num 89.6%

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

      5. associate-*r* 89.6%

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

      6. clear-num 89.5%

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

      7. un-div-inv 89.6%

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

      8. sub-neg 89.6%

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

      9. distribute-rgt-in 89.6%

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

      10. *-un-lft-identity 89.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 38.7%

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

      13. sqr-neg 38.7%

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

      14. sqrt-unprod 38.7%

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

      15. add-sqr-sqrt 38.7%

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

   8. Applied egg-rr 38.7%

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

   9. Taylor expanded in m around 0 38.8%

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

   if 4.1999999999999998e-188 < v

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      16. 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 42.8%

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

      1. neg-mul-1 42.8%

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

   6. Simplified 42.8%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 4.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 12: 27.1% 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 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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. distribute-lft-in 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-*l/ 93.9%

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

   6. associate-*r/ 99.8%

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

   7. *-lft-identity 99.8%

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

   8. associate-*l/ 99.8%

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

   9. associate-*r* 99.8%

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

   10. *-commutative 99.8%

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

   11. distribute-rgt-out 99.8%

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

   12. associate-*r/ 99.9%

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

   13. associate-/l* 99.9%

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

   14. /-rgt-identity 99.9%

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

   15. associate-*l/ 99.9%

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

   16. 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 30.0%

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

   1. neg-mul-1 30.0%

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

6. Simplified 30.0%

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

7. Final simplification 30.0%

   \[\leadsto -m \]

Reproduce

herbie shell --seed 2023230 
(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))