a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 9.0s

Alternatives: 9

Speedup: N/A×

• 41.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 99.8%

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

   1. neg-mul-1 99.8%

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

   2. +-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. div-sub 99.8%

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

   5. associate-*r/ 99.9%

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

   6. associate-*l/ 99.8%

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

   7. associate-/r/ 99.9%

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

7. Simplified 99.9%

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

Alternative 2: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.05 \cdot 10^{-131}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 - \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.05e-131)
   (- m)
   (if (<= m 1.0) (* m (/ m v)) (* m (- -1.0 (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.05e-131) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = m * (-1.0 - (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.05d-131) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m * (m / v)
    else
        tmp = m * ((-1.0d0) - (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.05e-131) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = m * (-1.0 - (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.05e-131:
		tmp = -m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = m * (-1.0 - (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.05e-131)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(m * Float64(-1.0 - Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.05e-131)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = m * (-1.0 - (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.05e-131], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(-1.0 - N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.0500000000000001e-131

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. +-commutative 99.8%

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

      3. sub-neg 99.8%

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

      4. div-sub 99.8%

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

      5. associate-*r/ 99.8%

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

      6. associate-*l/ 99.8%

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

      7. associate-/r/ 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in m around 0 70.8%

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

      1. neg-mul-1 70.8%

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

   10. Simplified 70.8%

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

   if 2.0500000000000001e-131 < 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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. clear-num 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in m around 0 95.6%

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

   8. Taylor expanded in v around 0 62.8%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. associate-/r/ 0.1%

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

      2. fma-define 0.1%

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

      3. frac-2neg 0.1%

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

      4. metadata-eval 0.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 80.9%

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

      7. sqr-neg 80.9%

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

      8. sqrt-unprod 81.0%

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

      9. add-sqr-sqrt 81.0%

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

   9. Applied egg-rr 81.0%

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

       1. fma-undefine 81.0%

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

       2. associate-*l/ 81.0%

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

       3. neg-mul-1 81.0%

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

       4. distribute-neg-frac 81.0%

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

       5. metadata-eval 81.0%

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

       6. distribute-neg-in 81.0%

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

       7. +-commutative 81.0%

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

       8. distribute-neg-in 81.0%

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

       9. metadata-eval 81.0%

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

       10. sub-neg 81.0%

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

   11. Simplified 81.0%

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

Alternative 3: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-18}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2e-18) (- (* m (/ m v)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2e-18) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2d-18) then
        tmp = (m * (m / v)) - m
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2e-18) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2e-18:
		tmp = (m * (m / v)) - m
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2e-18)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2e-18)
		tmp = (m * (m / v)) - m;
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2e-18], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.0000000000000001e-18

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. clear-num 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in m around 0 99.7%

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

      1. distribute-rgt-in 99.7%

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

      2. clear-num 99.8%

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

      3. div-inv 99.8%

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

      4. associate-*r* 99.8%

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

      5. associate-/r/ 99.7%

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

      6. clear-num 99.8%

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

      7. neg-mul-1 99.8%

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

   9. Applied egg-rr 99.8%

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

   if 2.0000000000000001e-18 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*l* 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-18}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (m * (m / v)) - m;
	else
		tmp = m * (-1.0 - (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[(-1.0 - N[(m / v), $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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. clear-num 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in m around 0 97.7%

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

      1. distribute-rgt-in 97.7%

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

      2. clear-num 97.9%

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

      3. div-inv 97.8%

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

      4. associate-*r* 97.8%

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

      5. associate-/r/ 97.7%

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

      6. clear-num 97.9%

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

      7. neg-mul-1 97.9%

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

   9. Applied egg-rr 97.9%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. associate-/r/ 0.1%

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

      2. fma-define 0.1%

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

      3. frac-2neg 0.1%

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

      4. metadata-eval 0.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 80.9%

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

      7. sqr-neg 80.9%

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

      8. sqrt-unprod 81.0%

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

      9. add-sqr-sqrt 81.0%

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

   9. Applied egg-rr 81.0%

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

       1. fma-undefine 81.0%

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

       2. associate-*l/ 81.0%

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

       3. neg-mul-1 81.0%

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

       4. distribute-neg-frac 81.0%

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

       5. metadata-eval 81.0%

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

       6. distribute-neg-in 81.0%

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

       7. +-commutative 81.0%

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

       8. distribute-neg-in 81.0%

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

       9. metadata-eval 81.0%

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

       10. sub-neg 81.0%

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

   11. Simplified 81.0%

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

4. Final simplification 88.6%

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

Alternative 5: 87.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m * (-1.0 - (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(-1.0 - 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 * (-1.0 - (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[(-1.0 - N[(m / v), $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. Add Preprocessing

   3. Taylor expanded in m around 0 97.9%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. associate-/r/ 0.1%

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

      2. fma-define 0.1%

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

      3. frac-2neg 0.1%

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

      4. metadata-eval 0.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 80.9%

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

      7. sqr-neg 80.9%

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

      8. sqrt-unprod 81.0%

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

      9. add-sqr-sqrt 81.0%

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

   9. Applied egg-rr 81.0%

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

       1. fma-undefine 81.0%

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

       2. associate-*l/ 81.0%

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

       3. neg-mul-1 81.0%

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

       4. distribute-neg-frac 81.0%

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

       5. metadata-eval 81.0%

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

       6. distribute-neg-in 81.0%

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

       7. +-commutative 81.0%

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

       8. distribute-neg-in 81.0%

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

       9. metadata-eval 81.0%

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

       10. sub-neg 81.0%

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

   11. Simplified 81.0%

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

4. Final simplification 88.6%

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

Alternative 6: 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 7: 37.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 8 \cdot 10^{-132}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 8e-132) (* m (/ m v)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 8e-132) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 8d-132) then
        tmp = m * (m / v)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 8e-132) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 8e-132:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 8e-132)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 8e-132)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 8e-132], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 7.9999999999999999e-132

   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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in m around 0 42.0%

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

   8. Taylor expanded in v around 0 32.6%

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

   if 7.9999999999999999e-132 < v

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 99.9%

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

      1. neg-mul-1 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-*l/ 99.9%

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

      7. associate-/r/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around 0 40.9%

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

      1. neg-mul-1 40.9%

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

   10. Simplified 40.9%

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

Alternative 8: 26.9% 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 99.8%

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

   1. neg-mul-1 99.8%

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

   2. +-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. div-sub 99.8%

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

   5. associate-*r/ 99.9%

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

   6. associate-*l/ 99.8%

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

   7. associate-/r/ 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in m around 0 25.7%

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

   1. neg-mul-1 25.7%

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

10. Simplified 25.7%

    \[\leadsto \color{blue}{-m} \]
11. Add Preprocessing

Alternative 9: 3.1% 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 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 25.7%

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

   1. *-commutative 25.7%

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

   2. neg-mul-1 25.7%

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

   3. neg-sub0 25.7%

      \[\leadsto \color{blue}{0 - m} \]

   4. sub-neg 25.7%

      \[\leadsto \color{blue}{0 + \left(-m\right)} \]

   5. add-sqr-sqrt 0.0%

      \[\leadsto 0 + \color{blue}{\sqrt{-m} \cdot \sqrt{-m}} \]

   6. sqrt-unprod 2.6%

      \[\leadsto 0 + \color{blue}{\sqrt{\left(-m\right) \cdot \left(-m\right)}} \]

   7. sqr-neg 2.6%

      \[\leadsto 0 + \sqrt{\color{blue}{m \cdot m}} \]

   8. sqrt-unprod 2.6%

      \[\leadsto 0 + \color{blue}{\sqrt{m} \cdot \sqrt{m}} \]

   9. add-sqr-sqrt 2.6%

      \[\leadsto 0 + \color{blue}{m} \]

7. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{0 + m} \]
8. Step-by-step derivation

   1. +-lft-identity 2.6%

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

9. Simplified 2.6%

   \[\leadsto \color{blue}{m} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024182 
(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))