a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

Alternatives: 14

Speedup: 1.0×

• 41.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-220}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-141}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-118}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7.2e-220)
   (- m)
   (if (<= m 5e-141)
     (* m (/ m v))
     (if (<= m 4.8e-118)
       (- m)
       (if (<= m 1.0) (/ (* m m) v) (* m (/ (- m) v)))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.2e-220) {
		tmp = -m;
	} else if (m <= 5e-141) {
		tmp = m * (m / v);
	} else if (m <= 4.8e-118) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = (m * m) / v;
	} else {
		tmp = m * (-m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 7.2d-220) then
        tmp = -m
    else if (m <= 5d-141) then
        tmp = m * (m / v)
    else if (m <= 4.8d-118) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = (m * m) / v
    else
        tmp = m * (-m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7.2e-220) {
		tmp = -m;
	} else if (m <= 5e-141) {
		tmp = m * (m / v);
	} else if (m <= 4.8e-118) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = (m * m) / v;
	} else {
		tmp = m * (-m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.2e-220:
		tmp = -m
	elif m <= 5e-141:
		tmp = m * (m / v)
	elif m <= 4.8e-118:
		tmp = -m
	elif m <= 1.0:
		tmp = (m * m) / v
	else:
		tmp = m * (-m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.2e-220)
		tmp = Float64(-m);
	elseif (m <= 5e-141)
		tmp = Float64(m * Float64(m / v));
	elseif (m <= 4.8e-118)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(Float64(m * 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 <= 7.2e-220)
		tmp = -m;
	elseif (m <= 5e-141)
		tmp = m * (m / v);
	elseif (m <= 4.8e-118)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = (m * m) / v;
	else
		tmp = m * (-m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.2e-220], (-m), If[LessEqual[m, 5e-141], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.8e-118], (-m), If[LessEqual[m, 1.0], N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision], N[(m * N[((-m) / v), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if m < 7.20000000000000042e-220 or 4.9999999999999999e-141 < m < 4.8000000000000003e-118

   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/ 88.0%

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

      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 m around 0 84.6%

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

      1. neg-mul-1 84.6%

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

   6. Simplified 84.6%

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

   if 7.20000000000000042e-220 < m < 4.9999999999999999e-141

   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/ 49.1%

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

      6. associate-*r/ 99.6%

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

      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/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      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/ 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in v around 0 18.0%

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

      1. associate-/l* 18.0%

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

      2. unpow2 18.0%

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

   6. Simplified 18.0%

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

   7. Taylor expanded in m around 0 18.0%

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

      1. unpow2 18.0%

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

      2. associate-*r/ 65.8%

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

   9. Simplified 65.8%

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

   if 4.8000000000000003e-118 < 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/ 99.8%

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

      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/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      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/ 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in v around 0 77.1%

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

      1. associate-/l* 77.1%

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

      2. unpow2 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in m around 0 70.1%

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

      1. unpow2 70.1%

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

   9. Simplified 70.1%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 99.8%

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

      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 99.8%

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

      1. associate-/l* 99.8%

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

      2. unpow2 99.8%

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

   6. Simplified 99.8%

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

   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. frac-2neg 0.1%

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

       2. associate-*r/ 0.1%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 76.0%

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

       5. sqr-neg 76.0%

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

       6. sqrt-unprod 76.0%

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

       7. add-sqr-sqrt 76.0%

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

   11. Applied egg-rr 76.0%

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

   12. Taylor expanded in m around 0 76.0%

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

       1. mul-1-neg 76.0%

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

       2. unpow2 76.0%

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

       3. associate-*l/ 76.0%

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

       4. distribute-rgt-neg-out 76.0%

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

   14. Simplified 76.0%

       \[\leadsto \color{blue}{\frac{m}{v} \cdot \left(-m\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-220}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-141}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-118}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.05e-21], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m * m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.05000000000000006e-21

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. 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/ 79.3%

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

      6. associate-*r/ 99.8%

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

      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/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      15. associate-/l* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in m around 0 79.3%

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

      1. mul-1-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. unsub-neg 79.3%

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

      4. unpow2 79.3%

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

      5. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

   if 1.05000000000000006e-21 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 99.8%

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

      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 99.8%

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

      1. associate-*r/ 99.7%

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

      2. unpow2 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 4.99999999999999954e-22

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. 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/ 79.3%

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

      6. associate-*r/ 99.8%

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

      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/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      15. associate-/l* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in m around 0 79.3%

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

      1. mul-1-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. unsub-neg 79.3%

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

      4. unpow2 79.3%

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

      5. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

   if 4.99999999999999954e-22 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 99.8%

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

      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 99.8%

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

      1. associate-/l* 99.7%

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

      2. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   8. Applied egg-rr 99.7%

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

      1. associate-/l* 99.7%

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

      2. associate-/r/ 99.7%

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

      3. un-div-inv 99.7%

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

      4. associate-*l/ 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. clear-num 99.7%

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

      7. associate-*r/ 99.7%

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.05e-21) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = (m * 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 <= 1.05d-21) then
        tmp = (m * (m / v)) - m
    else
        tmp = (m * m) / (v / (1.0d0 - m))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

Wolfram

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

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. 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/ 79.3%

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

      6. associate-*r/ 99.8%

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

      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/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      15. associate-/l* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in m around 0 79.3%

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

      1. mul-1-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. unsub-neg 79.3%

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

      4. unpow2 79.3%

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

      5. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

   if 1.05000000000000006e-21 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 99.8%

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

      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 99.8%

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

      1. associate-/l* 99.7%

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

      2. unpow2 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.05e-21], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.05000000000000006e-21

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. 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/ 79.3%

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

      6. associate-*r/ 99.8%

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

      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/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      15. associate-/l* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in m around 0 79.3%

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

      1. mul-1-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. unsub-neg 79.3%

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

      4. unpow2 79.3%

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

      5. associate-*r/ 99.8%

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

   8. Simplified 99.8%

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

   if 1.05000000000000006e-21 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 99.8%

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

      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 99.8%

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

      1. associate-*r/ 99.7%

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

      2. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. associate-*l/ 99.7%

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

      4. div-inv 99.7%

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

      5. times-frac 99.6%

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

      6. associate-*r/ 99.7%

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

      7. div-inv 99.7%

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

      8. clear-num 99.8%

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

      9. /-rgt-identity 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 7: 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 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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/ 90.6%

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

   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/ 99.8%

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

   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. Final simplification 99.8%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
2. 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/ 90.6%

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

   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/ 99.8%

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

   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%

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

   16. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 9: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 81.1%

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

      6. associate-*r/ 99.7%

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

      7. *-lft-identity 99.7%

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

      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/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      15. associate-/l* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in m around 0 78.6%

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

      1. mul-1-neg 78.6%

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

      2. +-commutative 78.6%

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

      3. unsub-neg 78.6%

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

      4. unpow2 78.6%

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

      5. associate-*r/ 97.3%

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

   8. Simplified 97.3%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 99.8%

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

      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 99.8%

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

      1. associate-*r/ 99.7%

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

      2. unpow2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in m around inf 98.3%

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

      1. neg-mul-1 98.3%

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

      2. distribute-neg-frac 98.3%

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

   9. Simplified 98.3%

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

4. Final simplification 97.8%

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

Alternative 10: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 81.1%

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

      6. associate-*r/ 99.7%

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

      7. *-lft-identity 99.7%

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

      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/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      15. associate-/l* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in m around 0 78.6%

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

      1. mul-1-neg 78.6%

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

      2. +-commutative 78.6%

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

      3. unsub-neg 78.6%

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

      4. unpow2 78.6%

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

      5. associate-*r/ 97.3%

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

   8. Simplified 97.3%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 99.8%

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

      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 99.8%

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

      1. associate-/l* 99.8%

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

      2. unpow2 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in m around inf 98.3%

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

      1. associate-*r/ 98.3%

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

      2. neg-mul-1 98.3%

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

   9. Simplified 98.3%

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

4. Final simplification 97.8%

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

Alternative 11: 87.6% accurate, 1.2× speedup

Math

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

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 99.7%

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

   2. Taylor expanded in m around 0 97.2%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 99.8%

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

      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 99.8%

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

      1. associate-/l* 99.8%

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

      2. unpow2 99.8%

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

   6. Simplified 99.8%

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

   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. frac-2neg 0.1%

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

       2. associate-*r/ 0.1%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 76.0%

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

       5. sqr-neg 76.0%

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

       6. sqrt-unprod 76.0%

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

       7. add-sqr-sqrt 76.0%

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

   11. Applied egg-rr 76.0%

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

   12. Taylor expanded in m around 0 76.0%

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

       1. mul-1-neg 76.0%

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

       2. unpow2 76.0%

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

       3. associate-*l/ 76.0%

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

       4. distribute-rgt-neg-out 76.0%

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

   14. Simplified 76.0%

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

4. Final simplification 86.5%

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

Alternative 12: 87.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 81.1%

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

      6. associate-*r/ 99.7%

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

      7. *-lft-identity 99.7%

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

      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/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      15. associate-/l* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in m around 0 78.6%

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

      1. mul-1-neg 78.6%

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

      2. +-commutative 78.6%

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

      3. unsub-neg 78.6%

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

      4. unpow2 78.6%

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

      5. associate-*r/ 97.3%

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

   8. Simplified 97.3%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 99.8%

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

      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 99.8%

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

      1. associate-/l* 99.8%

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

      2. unpow2 99.8%

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

   6. Simplified 99.8%

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

   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. frac-2neg 0.1%

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

       2. associate-*r/ 0.1%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 76.0%

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

       5. sqr-neg 76.0%

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

       6. sqrt-unprod 76.0%

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

       7. add-sqr-sqrt 76.0%

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

   11. Applied egg-rr 76.0%

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

   12. Taylor expanded in m around 0 76.0%

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

       1. mul-1-neg 76.0%

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

       2. unpow2 76.0%

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

       3. associate-*l/ 76.0%

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

       4. distribute-rgt-neg-out 76.0%

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

   14. Simplified 76.0%

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

4. Final simplification 86.5%

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

Alternative 13: 38.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 1.9e-148) (* m (/ m v)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 1.9e-148) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 1.9d-148) then
        tmp = m * (m / v)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 1.9e-148) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 1.9e-148:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 1.9e-148)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 1.9e-148)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 1.9e-148], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 1.90000000000000007e-148

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 81.6%

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

      6. associate-*r/ 99.7%

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

      7. *-lft-identity 99.7%

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

      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/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      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 70.8%

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

      1. associate-/l* 70.7%

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

      2. unpow2 70.7%

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

   6. Simplified 70.7%

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

   7. Taylor expanded in m around 0 22.8%

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

      1. unpow2 22.8%

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

      2. associate-*r/ 38.9%

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

   9. Simplified 38.9%

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

   if 1.90000000000000007e-148 < 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/ 99.9%

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

      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/ 99.8%

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

      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 38.4%

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

      1. neg-mul-1 38.4%

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

   6. Simplified 38.4%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 14: 27.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ -m \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (- m))

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = -m
end function

Java

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

Python

def code(m, v):
	return -m

Julia

function code(m, v)
	return Float64(-m)
end

MATLAB

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

Wolfram

code[m_, v_] := (-m)

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. 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/ 90.6%

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

   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/ 99.8%

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

   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 26.9%

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

   1. neg-mul-1 26.9%

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

6. Simplified 26.9%

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

7. Final simplification 26.9%

   \[\leadsto -m \]

Reproduce

herbie shell --seed 2023213 
(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))