a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 8.4s

Alternatives: 9

Speedup: 1.0×

• 41.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.3 \cdot 10^{-149}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 6.8 \cdot 10^{-133}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-123}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{-m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.3e-149)
   (- m)
   (if (<= m 6.8e-133)
     (* m (/ m v))
     (if (<= m 8e-123)
       (- m)
       (if (<= m 1.0) (/ m (/ v m)) (/ m (/ v (- m))))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.3e-149) {
		tmp = -m;
	} else if (m <= 6.8e-133) {
		tmp = m * (m / v);
	} else if (m <= 8e-123) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = 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 <= 6.3d-149) then
        tmp = -m
    else if (m <= 6.8d-133) then
        tmp = m * (m / v)
    else if (m <= 8d-123) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m / (v / m)
    else
        tmp = m / (v / -m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 6.3e-149) {
		tmp = -m;
	} else if (m <= 6.8e-133) {
		tmp = m * (m / v);
	} else if (m <= 8e-123) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = m / (v / -m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.3e-149:
		tmp = -m
	elif m <= 6.8e-133:
		tmp = m * (m / v)
	elif m <= 8e-123:
		tmp = -m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = m / (v / -m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.3e-149)
		tmp = Float64(-m);
	elseif (m <= 6.8e-133)
		tmp = Float64(m * Float64(m / v));
	elseif (m <= 8e-123)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(m / Float64(v / Float64(-m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.3e-149)
		tmp = -m;
	elseif (m <= 6.8e-133)
		tmp = m * (m / v);
	elseif (m <= 8e-123)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = m / (v / -m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.3e-149], (-m), If[LessEqual[m, 6.8e-133], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 8e-123], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / (-m)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if m < 6.29999999999999989e-149 or 6.80000000000000012e-133 < m < 8.0000000000000005e-123

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 75.8%

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

      1. neg-mul-1 75.8%

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

   7. Simplified 75.8%

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

   if 6.29999999999999989e-149 < m < 6.80000000000000012e-133

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.5%

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

      4. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in v around 0 88.7%

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

   6. Taylor expanded in m around 0 88.7%

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

      1. unpow2 88.7%

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

      2. associate-*l/ 88.9%

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

   8. Applied egg-rr 88.9%

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

   if 8.0000000000000005e-123 < m < 1

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-/l* 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in v around 0 85.0%

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

      1. *-commutative 85.0%

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

      2. unpow2 85.0%

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

      3. associate-*r* 85.0%

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

      4. associate-*l/ 84.9%

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

      5. associate-*r/ 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. clear-num 84.8%

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

      2. div-inv 84.8%

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

      3. *-commutative 84.8%

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

      4. clear-num 84.8%

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

      5. un-div-inv 85.0%

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

      6. associate-/l/ 85.0%

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

      7. *-commutative 85.0%

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

   9. Applied egg-rr 85.0%

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

   10. Taylor expanded in m around 0 82.6%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 100.0%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. unpow2 0.1%

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

      2. associate-*l/ 0.1%

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

   8. Applied egg-rr 0.1%

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

      1. clear-num 0.1%

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

      2. associate-*l/ 0.1%

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

      3. *-un-lft-identity 0.1%

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

      4. div-inv 0.1%

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

      5. frac-2neg 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 81.5%

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

      8. sqr-neg 81.5%

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

      9. sqrt-prod 81.5%

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

      10. add-sqr-sqrt 81.5%

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

      11. div-inv 81.5%

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

   10. Applied egg-rr 81.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.3 \cdot 10^{-149}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 6.8 \cdot 10^{-133}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-123}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{-m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.2 \cdot 10^{-150} \lor \neg \left(m \leq 7.8 \cdot 10^{-133}\right) \land \left(m \leq 7.5 \cdot 10^{-123} \lor \neg \left(m \leq 1\right)\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 9.2e-150)
         (and (not (<= m 7.8e-133)) (or (<= m 7.5e-123) (not (<= m 1.0)))))
   (- m)
   (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 9.2e-150) || (!(m <= 7.8e-133) && ((m <= 7.5e-123) || !(m <= 1.0)))) {
		tmp = -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 <= 9.2d-150) .or. (.not. (m <= 7.8d-133)) .and. (m <= 7.5d-123) .or. (.not. (m <= 1.0d0))) then
        tmp = -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 <= 9.2e-150) || (!(m <= 7.8e-133) && ((m <= 7.5e-123) || !(m <= 1.0)))) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 9.2e-150) or (not (m <= 7.8e-133) and ((m <= 7.5e-123) or not (m <= 1.0))):
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((m <= 9.2e-150) || (!(m <= 7.8e-133) && ((m <= 7.5e-123) || !(m <= 1.0))))
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m <= 9.2e-150) || (~((m <= 7.8e-133)) && ((m <= 7.5e-123) || ~((m <= 1.0)))))
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 9.2e-150], And[N[Not[LessEqual[m, 7.8e-133]], $MachinePrecision], Or[LessEqual[m, 7.5e-123], N[Not[LessEqual[m, 1.0]], $MachinePrecision]]]], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.20000000000000011e-150 or 7.80000000000000058e-133 < m < 7.50000000000000011e-123 or 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 33.1%

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

      1. neg-mul-1 33.1%

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

   7. Simplified 33.1%

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

   if 9.20000000000000011e-150 < m < 7.80000000000000058e-133 or 7.50000000000000011e-123 < 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. associate-/l* 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in v around 0 85.5%

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

   6. Taylor expanded in m around 0 83.5%

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

      1. unpow2 83.5%

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

      2. associate-*l/ 83.4%

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

   8. Applied egg-rr 83.4%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.2 \cdot 10^{-150} \lor \neg \left(m \leq 7.8 \cdot 10^{-133}\right) \land \left(m \leq 7.5 \cdot 10^{-123} \lor \neg \left(m \leq 1\right)\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.2 \cdot 10^{-150}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 5.8 \cdot 10^{-133}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 1.35 \cdot 10^{-122} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.2e-150)
   (- m)
   (if (<= m 5.8e-133)
     (* m (/ m v))
     (if (or (<= m 1.35e-122) (not (<= m 1.0))) (- m) (/ m (/ v m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.2e-150) {
		tmp = -m;
	} else if (m <= 5.8e-133) {
		tmp = m * (m / v);
	} else if ((m <= 1.35e-122) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = 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 <= 6.2d-150) then
        tmp = -m
    else if (m <= 5.8d-133) then
        tmp = m * (m / v)
    else if ((m <= 1.35d-122) .or. (.not. (m <= 1.0d0))) then
        tmp = -m
    else
        tmp = m / (v / m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 6.2e-150) {
		tmp = -m;
	} else if (m <= 5.8e-133) {
		tmp = m * (m / v);
	} else if ((m <= 1.35e-122) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m / (v / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.2e-150:
		tmp = -m
	elif m <= 5.8e-133:
		tmp = m * (m / v)
	elif (m <= 1.35e-122) or not (m <= 1.0):
		tmp = -m
	else:
		tmp = m / (v / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.2e-150)
		tmp = Float64(-m);
	elseif (m <= 5.8e-133)
		tmp = Float64(m * Float64(m / v));
	elseif ((m <= 1.35e-122) || !(m <= 1.0))
		tmp = Float64(-m);
	else
		tmp = Float64(m / Float64(v / m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.2e-150)
		tmp = -m;
	elseif (m <= 5.8e-133)
		tmp = m * (m / v);
	elseif ((m <= 1.35e-122) || ~((m <= 1.0)))
		tmp = -m;
	else
		tmp = m / (v / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.2e-150], (-m), If[LessEqual[m, 5.8e-133], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[m, 1.35e-122], N[Not[LessEqual[m, 1.0]], $MachinePrecision]], (-m), N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < 6.19999999999999996e-150 or 5.7999999999999997e-133 < m < 1.35000000000000005e-122 or 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 33.1%

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

      1. neg-mul-1 33.1%

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

   7. Simplified 33.1%

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

   if 6.19999999999999996e-150 < m < 5.7999999999999997e-133

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.5%

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

      4. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in v around 0 88.7%

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

   6. Taylor expanded in m around 0 88.7%

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

      1. unpow2 88.7%

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

      2. associate-*l/ 88.9%

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

   8. Applied egg-rr 88.9%

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

   if 1.35000000000000005e-122 < m < 1

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-/l* 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in v around 0 85.0%

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

      1. *-commutative 85.0%

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

      2. unpow2 85.0%

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

      3. associate-*r* 85.0%

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

      4. associate-*l/ 84.9%

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

      5. associate-*r/ 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. clear-num 84.8%

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

      2. div-inv 84.8%

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

      3. *-commutative 84.8%

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

      4. clear-num 84.8%

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

      5. un-div-inv 85.0%

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

      6. associate-/l/ 85.0%

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

      7. *-commutative 85.0%

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

   9. Applied egg-rr 85.0%

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

   10. Taylor expanded in m around 0 82.6%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.2 \cdot 10^{-150}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 5.8 \cdot 10^{-133}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 1.35 \cdot 10^{-122} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.8e-16) (* m (+ -1.0 (/ m v))) (* m (* (- 1.0 m) (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.8e-16) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * ((1.0 - m) * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.8d-16) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = m * ((1.0d0 - m) * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.8e-16) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * ((1.0 - m) * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.8e-16:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m * ((1.0 - m) * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.8e-16)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(Float64(1.0 - m) * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.8e-16)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = m * ((1.0 - m) * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.8e-16], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.8000000000000001e-16

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0 99.7%

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

   if 2.8000000000000001e-16 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*l/ 100.0%

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

      5. associate-*r/ 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0) (* m (+ -1.0 (/ m v))) (* m (/ (- (- m) v) 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) / 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) / 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) / 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) / 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(Float64(-m) - v) / 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) / 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[(N[((-m) - v), $MachinePrecision] / 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. Add Preprocessing

   3. Taylor expanded in m around 0 98.8%

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

   if 1 < m

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 0.1%

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

   4. Taylor expanded in v around 0 0.1%

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

      1. associate-*r* 0.1%

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

      2. neg-mul-1 0.1%

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

      3. unpow2 0.1%

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

      4. sqr-neg 0.1%

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

      5. distribute-lft-out 0.1%

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

      6. unsub-neg 0.1%

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

   6. Simplified 0.1%

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

      1. frac-2neg 0.1%

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

      2. div-inv 0.1%

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

      3. distribute-lft-neg-out 0.1%

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

      4. remove-double-neg 0.1%

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

      5. sub-neg 0.1%

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

      6. +-commutative 0.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 81.5%

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

      9. sqr-neg 81.5%

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

      10. sqrt-prod 81.5%

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

      11. add-sqr-sqrt 81.5%

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

   8. Applied egg-rr 81.5%

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

      1. associate-*l* 81.5%

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

      2. associate-*r/ 81.5%

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

      3. *-commutative 81.5%

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

      4. *-lft-identity 81.5%

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

   10. Simplified 81.5%

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

4. Final simplification 90.9%

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

Alternative 7: 87.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   3. Taylor expanded in m around 0 98.8%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 100.0%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. unpow2 0.1%

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

      2. associate-*l/ 0.1%

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

   8. Applied egg-rr 0.1%

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

      1. clear-num 0.1%

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

      2. associate-*l/ 0.1%

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

      3. *-un-lft-identity 0.1%

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

      4. div-inv 0.1%

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

      5. frac-2neg 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 81.5%

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

      8. sqr-neg 81.5%

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

      9. sqrt-prod 81.5%

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

      10. add-sqr-sqrt 81.5%

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

      11. div-inv 81.5%

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

   10. Applied egg-rr 81.5%

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

4. Final simplification 90.9%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 27.7% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 28.2%

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

   1. neg-mul-1 28.2%

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

7. Simplified 28.2%

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

8. Final simplification 28.2%

   \[\leadsto -m \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(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))