a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 6.6s

Alternatives: 14

Speedup: 1.0×

• 42.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% 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 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.6 \cdot 10^{-54}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 8.6e-54) (- (* m (/ m v)) m) (* m (* (- 1.0 m) (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.6e-54) {
		tmp = (m * (m / v)) - m;
	} 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 <= 8.6d-54) then
        tmp = (m * (m / v)) - m
    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 <= 8.6e-54) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * ((1.0 - m) * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8.6e-54:
		tmp = (m * (m / v)) - m
	else:
		tmp = m * ((1.0 - m) * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8.6e-54)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	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 <= 8.6e-54)
		tmp = (m * (m / v)) - m;
	else
		tmp = m * ((1.0 - m) * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 8.6e-54], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $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 < 8.5999999999999999e-54

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.9%

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

      2. *-commutative 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. div-inv 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l* 99.8%

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

      8. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. unpow2 84.7%

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

      3. associate-*r/ 99.9%

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

      4. mul-1-neg 99.9%

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

      5. unsub-neg 99.9%

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

   8. Simplified 99.9%

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

   if 8.5999999999999999e-54 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.2%

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

      1. unpow2 99.2%

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

      2. associate-*r* 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-*r/ 99.2%

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

      5. associate-*r* 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.6 \cdot 10^{-54}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.6 \cdot 10^{-54}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 8.6e-54) (- (* m (/ m v)) m) (* m (/ m (/ v (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.6e-54) {
		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 <= 8.6d-54) 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 <= 8.6e-54) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * (m / (v / (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8.6e-54:
		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 <= 8.6e-54)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(m * Float64(m / Float64(v / Float64(1.0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 8.6e-54)
		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, 8.6e-54], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.5999999999999999e-54

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.9%

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

      2. *-commutative 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. div-inv 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l* 99.8%

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

      8. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. unpow2 84.7%

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

      3. associate-*r/ 99.9%

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

      4. mul-1-neg 99.9%

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

      5. unsub-neg 99.9%

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

   8. Simplified 99.9%

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

   if 8.5999999999999999e-54 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.2%

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

      1. associate-/l* 99.2%

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

      2. unpow2 99.2%

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

      3. associate-*r/ 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.6 \cdot 10^{-54}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.6 \cdot 10^{-54}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 8.6e-54) (- (* m (/ m v)) m) (/ (* m m) (/ v (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.6e-54) {
		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 <= 8.6d-54) 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 <= 8.6e-54) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = (m * m) / (v / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8.6e-54:
		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 <= 8.6e-54)
		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 <= 8.6e-54)
		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, 8.6e-54], 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 < 8.5999999999999999e-54

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.9%

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

      2. *-commutative 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. div-inv 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l* 99.8%

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

      8. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. unpow2 84.7%

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

      3. associate-*r/ 99.9%

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

      4. mul-1-neg 99.9%

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

      5. unsub-neg 99.9%

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

   8. Simplified 99.9%

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

   if 8.5999999999999999e-54 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.2%

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

      1. associate-/l* 99.2%

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

      2. unpow2 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.6 \cdot 10^{-54}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 5: 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 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.7%

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

   2. associate-/r/ 99.8%

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

   3. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 7: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-177}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.25e-177)
   (- m)
   (if (<= m 5.2e-7) (* m (/ m v)) (* (/ m v) (- m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-177) {
		tmp = -m;
	} else if (m <= 5.2e-7) {
		tmp = m * (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 <= 1.25d-177) then
        tmp = -m
    else if (m <= 5.2d-7) then
        tmp = m * (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 <= 1.25e-177) {
		tmp = -m;
	} else if (m <= 5.2e-7) {
		tmp = m * (m / v);
	} else {
		tmp = (m / v) * -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.25e-177:
		tmp = -m
	elif m <= 5.2e-7:
		tmp = m * (m / v)
	else:
		tmp = (m / v) * -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.25e-177)
		tmp = Float64(-m);
	elseif (m <= 5.2e-7)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(Float64(m / v) * Float64(-m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.25e-177)
		tmp = -m;
	elseif (m <= 5.2e-7)
		tmp = m * (m / v);
	else
		tmp = (m / v) * -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.25e-177], (-m), If[LessEqual[m, 5.2e-7], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * (-m)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.25e-177

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 75.7%

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

      1. neg-mul-1 75.7%

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

   6. Simplified 75.7%

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

   if 1.25e-177 < m < 5.19999999999999998e-7

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in v around 0 73.5%

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

      1. unpow2 73.5%

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

      2. associate-*r* 73.5%

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

      3. *-commutative 73.5%

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

      4. associate-*r/ 76.6%

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

      5. associate-*r* 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in m around 0 71.6%

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

      1. unpow2 71.6%

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

      2. associate-*r/ 74.7%

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

   9. Simplified 74.7%

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

   if 5.19999999999999998e-7 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.2%

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

      1. unpow2 99.2%

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

      2. associate-*r* 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-*r/ 99.2%

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

      5. associate-*r* 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. unpow2 0.1%

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

      2. associate-*r/ 0.1%

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

   9. Simplified 0.1%

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

       1. associate-*r/ 0.1%

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

       2. add-sqr-sqrt 0.1%

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

       3. sqrt-unprod 0.1%

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

       4. sqr-neg 0.1%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 74.8%

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

       7. distribute-frac-neg 74.8%

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

       8. associate-*r/ 74.8%

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

   11. Applied egg-rr 74.8%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-177}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \]

Alternative 8: 97.8% accurate, 1.1× speedup

Math

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

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(m * Float64(-Float64(m / Float64(v / m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (m * (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[(m * (-N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.8%

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

      5. div-inv 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

      8. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in m around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. associate-*r/ 98.5%

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

      4. mul-1-neg 98.5%

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

      5. unsub-neg 98.5%

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

   8. Simplified 98.5%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. unpow2 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. unpow2 97.1%

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

      3. associate-/l* 97.1%

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

   9. Simplified 97.1%

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

4. Final simplification 97.8%

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

Alternative 9: 97.8% accurate, 1.1× speedup

Math

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

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(m * Float64(Float64(m / v) * Float64(-m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (m * (m / v)) - m;
	else
		tmp = m * ((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[(m * N[(N[(m / v), $MachinePrecision] * (-m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.8%

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

      5. div-inv 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

      8. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in m around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. associate-*r/ 98.5%

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

      4. mul-1-neg 98.5%

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

      5. unsub-neg 98.5%

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

   8. Simplified 98.5%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. unpow2 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. unpow2 97.1%

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

      3. associate-*l/ 97.1%

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

      4. distribute-rgt-neg-in 97.1%

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

   9. Simplified 97.1%

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

4. Final simplification 97.8%

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

Alternative 10: 97.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0) (- (* m (/ m v)) m) (* 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 * 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 * 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 * m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (m * (m / v)) - m
	else:
		tmp = m * (-(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(-Float64(m * 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 * 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[((-N[(m * m), $MachinePrecision]) / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.8%

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

      5. div-inv 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

      8. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in m around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. associate-*r/ 98.5%

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

      4. mul-1-neg 98.5%

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

      5. unsub-neg 98.5%

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

   8. Simplified 98.5%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. unpow2 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 97.1%

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

      1. unpow2 97.1%

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

      2. associate-*r/ 97.1%

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

      3. mul-1-neg 97.1%

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

   9. Simplified 97.1%

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

4. Final simplification 97.8%

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

Alternative 11: 37.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{-177}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.8e-177) (- m) (if (<= m 5.2e-7) (* m (/ m v)) (- m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.8e-177) {
		tmp = -m;
	} else if (m <= 5.2e-7) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.8d-177) then
        tmp = -m
    else if (m <= 5.2d-7) then
        tmp = m * (m / v)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.8e-177) {
		tmp = -m;
	} else if (m <= 5.2e-7) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.8e-177:
		tmp = -m
	elif m <= 5.2e-7:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.8e-177)
		tmp = Float64(-m);
	elseif (m <= 5.2e-7)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.8e-177)
		tmp = -m;
	elseif (m <= 5.2e-7)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.8e-177], (-m), If[LessEqual[m, 5.2e-7], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]]

Derivation

1. Split input into 2 regimes

2. if m < 2.79999999999999987e-177 or 5.19999999999999998e-7 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 27.0%

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

      1. neg-mul-1 27.0%

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

   6. Simplified 27.0%

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

   if 2.79999999999999987e-177 < m < 5.19999999999999998e-7

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in v around 0 73.5%

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

      1. unpow2 73.5%

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

      2. associate-*r* 73.5%

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

      3. *-commutative 73.5%

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

      4. associate-*r/ 76.6%

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

      5. associate-*r* 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in m around 0 71.6%

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

      1. unpow2 71.6%

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

      2. associate-*r/ 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{-177}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 12: 87.1% accurate, 1.2× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.7%

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

      3. fma-neg 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 98.5%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. unpow2 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. unpow2 0.1%

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

      2. associate-*r/ 0.1%

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

   9. Simplified 0.1%

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

       1. associate-*r/ 0.1%

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

       2. add-sqr-sqrt 0.1%

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

       3. sqrt-unprod 0.1%

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

       4. sqr-neg 0.1%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 75.3%

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

       7. distribute-frac-neg 75.3%

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

       8. associate-*r/ 75.3%

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

   11. Applied egg-rr 75.3%

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

4. Final simplification 86.8%

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

Alternative 13: 87.1% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0) (- (* m (/ m v)) 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 / 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 / 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 / v) * -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (m * (m / v)) - m
	else:
		tmp = (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 / v) * Float64(-m));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.8%

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

      5. div-inv 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

      8. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in m around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. associate-*r/ 98.5%

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

      4. mul-1-neg 98.5%

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

      5. unsub-neg 98.5%

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

   8. Simplified 98.5%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. unpow2 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. associate-*r* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. unpow2 0.1%

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

      2. associate-*r/ 0.1%

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

   9. Simplified 0.1%

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

       1. associate-*r/ 0.1%

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

       2. add-sqr-sqrt 0.1%

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

       3. sqrt-unprod 0.1%

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

       4. sqr-neg 0.1%

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

       5. sqrt-unprod 0.0%

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

       6. add-sqr-sqrt 75.3%

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

       7. distribute-frac-neg 75.3%

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

       8. associate-*r/ 75.3%

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

   11. Applied egg-rr 75.3%

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

4. Final simplification 86.8%

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

Alternative 14: 26.3% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 25.7%

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

   1. neg-mul-1 25.7%

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

6. Simplified 25.7%

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

7. Final simplification 25.7%

   \[\leadsto -m \]

Reproduce

herbie shell --seed 2023292 
(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))