a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 11.0s

Alternatives: 13

Speedup: 1.0×

• 42.2% 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%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.2e-20) (- (/ m (/ v m)) m) (* m (* (- 1.0 m) (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.2e-20) {
		tmp = (m / (v / m)) - 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 <= 1.2d-20) then
        tmp = (m / (v / m)) - 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 <= 1.2e-20) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * ((1.0 - m) * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.2e-20:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * ((1.0 - m) * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.2e-20)
		tmp = Float64(Float64(m / Float64(v / m)) - 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 <= 1.2e-20)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * ((1.0 - m) * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.2e-20], N[(N[(m / N[(v / m), $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 < 1.19999999999999996e-20

   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 m around 0 87.5%

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

      1. neg-mul-1 87.5%

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

      2. +-commutative 87.5%

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

      3. unsub-neg 87.5%

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

      4. unpow2 87.5%

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

      5. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

   if 1.19999999999999996e-20 < 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 99.8%

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

      1. unpow2 99.8%

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

      2. associate-*r* 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 99.8%

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

      5. associate-*r* 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{m}{\frac{v}{m}}\\ \mathbf{if}\;m \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;t_0 - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (/ m (/ v m)))) (if (<= m 3.5e-22) (- t_0 m) (* (- 1.0 m) t_0))))

C

double code(double m, double v) {
	double t_0 = m / (v / m);
	double tmp;
	if (m <= 3.5e-22) {
		tmp = t_0 - m;
	} else {
		tmp = (1.0 - m) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m / (v / m)
    if (m <= 3.5d-22) then
        tmp = t_0 - m
    else
        tmp = (1.0d0 - m) * t_0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = m / (v / m);
	double tmp;
	if (m <= 3.5e-22) {
		tmp = t_0 - m;
	} else {
		tmp = (1.0 - m) * t_0;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m / (v / m)
	tmp = 0
	if m <= 3.5e-22:
		tmp = t_0 - m
	else:
		tmp = (1.0 - m) * t_0
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m / Float64(v / m))
	tmp = 0.0
	if (m <= 3.5e-22)
		tmp = Float64(t_0 - m);
	else
		tmp = Float64(Float64(1.0 - m) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m / (v / m);
	tmp = 0.0;
	if (m <= 3.5e-22)
		tmp = t_0 - m;
	else
		tmp = (1.0 - m) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 3.5e-22], N[(t$95$0 - m), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 3.50000000000000005e-22

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. 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 m around 0 87.3%

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

      1. neg-mul-1 87.3%

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

      2. +-commutative 87.3%

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

      3. unsub-neg 87.3%

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

      4. unpow2 87.3%

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

      5. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

   if 3.50000000000000005e-22 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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 99.8%

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

      1. associate-*l/ 99.8%

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

      2. unpow2 99.8%

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

      3. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{m}}\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. 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. associate-/r/ 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 5: 75.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-148}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.9e-148) (- m) (if (<= m 1.0) (/ m (/ v m)) (* m (/ (- m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-148) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = m * (-m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.9d-148) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m / (v / m)
    else
        tmp = m * (-m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-148) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = m * (-m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.9e-148:
		tmp = -m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = m * (-m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.9e-148)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(m * Float64(Float64(-m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.9e-148)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = m * (-m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.9e-148], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(m * N[((-m) / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.8999999999999998e-148

   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 76.5%

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

      1. neg-mul-1 76.5%

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

   6. Simplified 76.5%

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

   if 2.8999999999999998e-148 < 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* 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in v around 0 79.7%

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

      1. unpow2 79.7%

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

      2. associate-*r* 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r/ 79.6%

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

      5. associate-*r* 79.6%

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

   6. Simplified 79.6%

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

   7. Taylor expanded in m around 0 77.2%

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

      1. unpow2 77.2%

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

      2. associate-/l* 77.2%

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

   9. Simplified 77.2%

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

   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 m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. unpow2 0.1%

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

      3. add-sqr-sqrt 0.1%

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

      4. sqrt-prod 0.1%

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

      5. sqr-neg 0.1%

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

      6. mul-1-neg 0.1%

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

      7. mul-1-neg 0.1%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 82.4%

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

      10. mul-1-neg 82.4%

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

      11. distribute-rgt-neg-in 82.4%

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

      12. unpow2 82.4%

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

      13. cancel-sign-sub-inv 82.4%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 29.2%

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

      16. mul-1-neg 29.2%

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

      17. mul-1-neg 29.2%

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

      18. sqr-neg 29.2%

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

      19. sqrt-prod 82.4%

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

      20. add-sqr-sqrt 82.4%

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

      21. unpow2 82.4%

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

      22. associate-*l* 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. associate-/l* 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in m around inf 82.4%

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

       1. mul-1-neg 82.4%

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

       2. unpow2 82.4%

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

       3. associate-*r/ 82.4%

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

       4. distribute-rgt-neg-in 82.4%

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

       5. distribute-neg-frac 82.4%

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

   11. Simplified 82.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-148}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \]

Alternative 6: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. 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 m around 0 86.7%

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

      1. neg-mul-1 86.7%

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

      2. +-commutative 86.7%

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

      3. unsub-neg 86.7%

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

      4. unpow2 86.7%

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

      5. associate-/l* 98.5%

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

   6. Simplified 98.5%

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

   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. associate-*l/ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

      1. associate-*l/ 99.9%

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

      2. associate-/l* 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in m around inf 98.0%

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

       1. mul-1-neg 98.0%

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

       2. unpow2 98.0%

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

       3. distribute-neg-frac 98.0%

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

   11. Simplified 98.0%

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

4. Final simplification 98.2%

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

Alternative 7: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. 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 m around 0 86.7%

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

      1. neg-mul-1 86.7%

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

      2. +-commutative 86.7%

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

      3. unsub-neg 86.7%

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

      4. unpow2 86.7%

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

      5. associate-/l* 98.5%

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

   6. Simplified 98.5%

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

   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. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 98.0%

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

      1. associate-*r/ 98.0%

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

      2. neg-mul-1 98.0%

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

   9. Simplified 98.0%

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

4. Final simplification 98.2%

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

Alternative 8: 38.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.3 \cdot 10^{-148}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.3e-148) (- m) (if (<= m 1.0) (* m (/ m v)) (- m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.3e-148) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 3.3d-148) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m * (m / v)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.3e-148) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.3e-148:
		tmp = -m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.3e-148)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.3e-148)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.3e-148], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]]

Derivation

1. Split input into 2 regimes

2. if m < 3.29999999999999974e-148 or 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 31.3%

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

      1. neg-mul-1 31.3%

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

   6. Simplified 31.3%

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

   if 3.29999999999999974e-148 < 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* 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in v around 0 79.7%

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

      1. unpow2 79.7%

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

      2. associate-*r* 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r/ 79.6%

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

      5. associate-*r* 79.6%

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

   6. Simplified 79.6%

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

   7. Taylor expanded in m around 0 77.2%

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

      1. unpow2 77.2%

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

      2. associate-*r/ 77.1%

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

   9. Simplified 77.1%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.3 \cdot 10^{-148}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 9: 38.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-148}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.65e-148) (- m) (if (<= m 1.0) (/ m (/ v m)) (- m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-148) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = -m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.65d-148) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m / (v / m)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-148) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.65e-148:
		tmp = -m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.65e-148)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.65e-148)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.65e-148], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], (-m)]]

Derivation

1. Split input into 2 regimes

2. if m < 2.64999999999999998e-148 or 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 31.3%

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

      1. neg-mul-1 31.3%

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

   6. Simplified 31.3%

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

   if 2.64999999999999998e-148 < 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* 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in v around 0 79.7%

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

      1. unpow2 79.7%

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

      2. associate-*r* 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r/ 79.6%

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

      5. associate-*r* 79.6%

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

   6. Simplified 79.6%

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

   7. Taylor expanded in m around 0 77.2%

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

      1. unpow2 77.2%

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

      2. associate-/l* 77.2%

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

   9. Simplified 77.2%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-148}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 10: 88.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. 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. associate-*r/ 99.6%

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

      3. fma-neg 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in m around 0 98.4%

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

   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 m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. unpow2 0.1%

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

      3. add-sqr-sqrt 0.1%

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

      4. sqrt-prod 0.1%

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

      5. sqr-neg 0.1%

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

      6. mul-1-neg 0.1%

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

      7. mul-1-neg 0.1%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 82.4%

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

      10. mul-1-neg 82.4%

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

      11. distribute-rgt-neg-in 82.4%

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

      12. unpow2 82.4%

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

      13. cancel-sign-sub-inv 82.4%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 29.2%

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

      16. mul-1-neg 29.2%

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

      17. mul-1-neg 29.2%

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

      18. sqr-neg 29.2%

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

      19. sqrt-prod 82.4%

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

      20. add-sqr-sqrt 82.4%

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

      21. unpow2 82.4%

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

      22. associate-*l* 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. associate-/l* 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in m around inf 82.4%

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

       1. mul-1-neg 82.4%

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

       2. unpow2 82.4%

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

       3. associate-*r/ 82.4%

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

       4. distribute-rgt-neg-in 82.4%

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

       5. distribute-neg-frac 82.4%

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

   11. Simplified 82.4%

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

4. Final simplification 90.6%

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

Alternative 11: 88.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. 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 m around 0 86.7%

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

      1. neg-mul-1 86.7%

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

      2. +-commutative 86.7%

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

      3. unsub-neg 86.7%

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

      4. unpow2 86.7%

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

      5. associate-/l* 98.5%

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

   6. Simplified 98.5%

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

   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 m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. unpow2 0.1%

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

      3. add-sqr-sqrt 0.1%

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

      4. sqrt-prod 0.1%

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

      5. sqr-neg 0.1%

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

      6. mul-1-neg 0.1%

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

      7. mul-1-neg 0.1%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 82.4%

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

      10. mul-1-neg 82.4%

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

      11. distribute-rgt-neg-in 82.4%

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

      12. unpow2 82.4%

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

      13. cancel-sign-sub-inv 82.4%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 29.2%

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

      16. mul-1-neg 29.2%

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

      17. mul-1-neg 29.2%

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

      18. sqr-neg 29.2%

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

      19. sqrt-prod 82.4%

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

      20. add-sqr-sqrt 82.4%

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

      21. unpow2 82.4%

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

      22. associate-*l* 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. associate-/l* 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in m around inf 82.4%

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

       1. mul-1-neg 82.4%

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

       2. unpow2 82.4%

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

       3. associate-*r/ 82.4%

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

       4. distribute-rgt-neg-in 82.4%

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

       5. distribute-neg-frac 82.4%

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

   11. Simplified 82.4%

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

4. Final simplification 90.7%

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

Alternative 12: 27.1% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in m around 0 28.7%

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

   1. neg-mul-1 28.7%

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

6. Simplified 28.7%

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

7. Final simplification 28.7%

   \[\leadsto -m \]

Alternative 13: 3.0% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (m v) :precision binary64 m)

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return m

Julia

function code(m, v)
	return m
end

MATLAB

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

Wolfram

code[m_, v_] := m

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in m around 0 44.7%

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

   1. div-inv 44.7%

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

   2. unpow2 44.7%

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

   3. add-sqr-sqrt 44.6%

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

   4. sqrt-prod 44.7%

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

   5. sqr-neg 44.7%

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

   6. mul-1-neg 44.7%

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

   7. mul-1-neg 44.7%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 65.7%

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

   10. mul-1-neg 65.7%

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

   11. distribute-rgt-neg-in 65.7%

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

   12. unpow2 65.7%

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

   13. cancel-sign-sub-inv 65.7%

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

   14. add-sqr-sqrt 0.0%

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

   15. sqrt-unprod 16.4%

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

   16. mul-1-neg 16.4%

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

   17. mul-1-neg 16.4%

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

   18. sqr-neg 16.4%

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

   19. sqrt-prod 42.1%

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

   20. add-sqr-sqrt 42.1%

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

   21. unpow2 42.1%

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

   22. associate-*l* 41.8%

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

6. Applied egg-rr 41.8%

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

   1. associate-*r/ 42.1%

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

   2. associate-/l* 41.8%

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

8. Simplified 41.8%

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

9. Taylor expanded in m around 0 3.2%

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

10. Final simplification 3.2%

    \[\leadsto m \]

Reproduce

herbie shell --seed 2023282 
(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))