a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 6.1s

Alternatives: 10

Speedup: 1.0×

• 42.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7.6e-14) (- (/ m (/ v m)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.6e-14) {
		tmp = (m / (v / m)) - 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 <= 7.6d-14) then
        tmp = (m / (v / m)) - 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 <= 7.6e-14) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.6e-14:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.6e-14)
		tmp = Float64(Float64(m / Float64(v / m)) - 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 <= 7.6e-14)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.6e-14], N[(N[(m / N[(v / m), $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 < 7.6000000000000004e-14

   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-*r/ 99.7%

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

      4. fma-def 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 81.1%

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

      1. neg-mul-1 81.1%

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

      2. +-commutative 81.1%

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

      3. unsub-neg 81.1%

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

      4. unpow2 81.1%

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

      5. associate-/l* 99.6%

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

   6. Simplified 99.6%

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

   if 7.6000000000000004e-14 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*r/ 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.6%

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

      1. associate-*r/ 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{1 - m}}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7.6e-14) (- (/ m (/ v m)) m) (/ m (/ (/ v (- 1.0 m)) m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.6e-14) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m / ((v / (1.0 - 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 <= 7.6d-14) then
        tmp = (m / (v / m)) - m
    else
        tmp = m / ((v / (1.0d0 - m)) / m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7.6e-14) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m / ((v / (1.0 - m)) / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.6e-14:
		tmp = (m / (v / m)) - m
	else:
		tmp = m / ((v / (1.0 - m)) / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.6e-14)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m / Float64(Float64(v / Float64(1.0 - m)) / m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7.6e-14)
		tmp = (m / (v / m)) - m;
	else
		tmp = m / ((v / (1.0 - m)) / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.6e-14], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m / N[(N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.6000000000000004e-14

   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-*r/ 99.7%

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

      4. fma-def 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 81.1%

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

      1. neg-mul-1 81.1%

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

      2. +-commutative 81.1%

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

      3. unsub-neg 81.1%

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

      4. unpow2 81.1%

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

      5. associate-/l* 99.6%

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

   6. Simplified 99.6%

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

   if 7.6000000000000004e-14 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*r/ 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.6%

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

      1. associate-*r/ 99.6%

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

   6. Simplified 99.6%

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

      1. associate-*r* 99.6%

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

      2. frac-2neg 99.6%

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

      3. associate-*r/ 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-/l* 99.6%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in v around 0 99.6%

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

       1. *-lft-identity 99.6%

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

       2. sub-neg 99.6%

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

       3. metadata-eval 99.6%

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

       4. associate-*l/ 99.6%

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

       5. +-commutative 99.6%

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

       6. metadata-eval 99.6%

          \[\leadsto \frac{m}{\frac{-1 \cdot \left(\frac{1}{\color{blue}{\left(0 - 1\right)} + m} \cdot v\right)}{m}} \]

       7. associate--r- 99.6%

          \[\leadsto \frac{m}{\frac{-1 \cdot \left(\frac{1}{\color{blue}{0 - \left(1 - m\right)}} \cdot v\right)}{m}} \]

       8. neg-sub0 99.6%

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

       9. neg-mul-1 99.6%

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

       10. associate-/r* 99.6%

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

       11. metadata-eval 99.6%

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

       12. associate-*l* 99.6%

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

       13. associate-*r/ 99.6%

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

       14. metadata-eval 99.6%

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

       15. associate-*l/ 99.6%

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

       16. *-lft-identity 99.6%

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

   13. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{1 - m}}{m}}\\ \end{array} \]

Alternative 4: 73.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.05 \cdot 10^{-122}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.05e-122) (- m) (if (<= m 1.0) (/ m (/ v m)) (* m (/ (- m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.05e-122) {
		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 <= 1.05d-122) 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 <= 1.05e-122) {
		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 <= 1.05e-122:
		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 <= 1.05e-122)
		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 <= 1.05e-122)
		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, 1.05e-122], (-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 < 1.04999999999999996e-122

   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-*r/ 99.8%

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

      4. fma-def 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 68.3%

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

      1. neg-mul-1 68.3%

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

   6. Simplified 68.3%

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

   if 1.04999999999999996e-122 < m < 1

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-*r/ 99.4%

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

      4. fma-def 99.4%

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

      5. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in v around 0 84.6%

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

      1. associate-*r/ 84.5%

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

   6. Simplified 84.5%

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

      1. associate-*r* 84.8%

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

      2. frac-2neg 84.8%

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

      3. associate-*r/ 84.8%

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

   8. Applied egg-rr 84.8%

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

      1. associate-/l* 84.8%

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

      2. associate-/l* 84.9%

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

      3. +-commutative 84.9%

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

   10. Simplified 84.9%

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

   11. Taylor expanded in m around 0 73.4%

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

       1. unpow2 73.4%

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

       2. associate-/l* 73.4%

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

   13. Simplified 73.4%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. frac-2neg 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 82.9%

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

      5. sqr-neg 82.9%

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

      6. sqrt-unprod 82.9%

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

      7. add-sqr-sqrt 82.9%

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

   9. Applied egg-rr 82.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.05 \cdot 10^{-122}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m * Float64(Float64(m * 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 * -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[(N[(m * (-m)), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. 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-*r/ 99.7%

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

      4. fma-def 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 78.0%

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

      1. neg-mul-1 78.0%

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

      2. +-commutative 78.0%

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

      3. unsub-neg 78.0%

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

      4. unpow2 78.0%

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

      5. associate-/l* 95.2%

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

   6. Simplified 95.2%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unpow2 99.3%

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

      4. distribute-rgt-neg-out 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 97.4%

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

Alternative 6: 86.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

   2. Taylor expanded in m around 0 95.2%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. frac-2neg 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 82.9%

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

      5. sqr-neg 82.9%

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

      6. sqrt-unprod 82.9%

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

      7. add-sqr-sqrt 82.9%

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

   9. Applied egg-rr 82.9%

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

4. Final simplification 88.8%

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

Alternative 7: 86.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 78.0%

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

      1. neg-mul-1 78.0%

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

      2. +-commutative 78.0%

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

      3. unsub-neg 78.0%

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

      4. unpow2 78.0%

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

      5. associate-/l* 95.2%

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

   6. Simplified 95.2%

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

      1. associate-/r/ 95.2%

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

   8. Applied egg-rr 95.2%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. frac-2neg 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 82.9%

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

      5. sqr-neg 82.9%

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

      6. sqrt-unprod 82.9%

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

      7. add-sqr-sqrt 82.9%

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

   9. Applied egg-rr 82.9%

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

4. Final simplification 88.8%

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

Alternative 8: 86.8% 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-*r/ 99.7%

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

      4. fma-def 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 78.0%

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

      1. neg-mul-1 78.0%

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

      2. +-commutative 78.0%

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

      3. unsub-neg 78.0%

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

      4. unpow2 78.0%

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

      5. associate-/l* 95.2%

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

   6. Simplified 95.2%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. frac-2neg 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 82.9%

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

      5. sqr-neg 82.9%

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

      6. sqrt-unprod 82.9%

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

      7. add-sqr-sqrt 82.9%

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

   9. Applied egg-rr 82.9%

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

4. Final simplification 88.8%

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

Alternative 9: 37.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 5.1 \cdot 10^{-143}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 5.1e-143) (* m (/ m v)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 5.1e-143) {
		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 <= 5.1d-143) 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 <= 5.1e-143) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 5.1e-143:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 5.1e-143)
		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 <= 5.1e-143)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 5.1e-143], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 5.1000000000000004e-143

   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-*r/ 99.8%

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

      4. fma-def 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in v around 0 89.6%

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

      1. associate-*r/ 89.6%

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

   6. Simplified 89.6%

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

      1. associate-*r* 74.0%

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

      2. frac-2neg 74.0%

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

      3. associate-*r/ 74.0%

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

   8. Applied egg-rr 74.0%

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

      1. associate-/l* 74.0%

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

      2. associate-/l* 89.6%

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

      3. +-commutative 89.6%

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

   10. Simplified 89.6%

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

   11. Taylor expanded in m around 0 19.5%

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

       1. unpow2 19.5%

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

       2. associate-*r/ 35.2%

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

   13. Simplified 35.2%

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

   if 5.1000000000000004e-143 < 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-*r/ 99.9%

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

      4. fma-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 37.8%

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

      1. neg-mul-1 37.8%

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

   6. Simplified 37.8%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 5.1 \cdot 10^{-143}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 10: 26.4% 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-*r/ 99.8%

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

   4. fma-def 99.8%

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

   5. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in m around 0 26.0%

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

   1. neg-mul-1 26.0%

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

6. Simplified 26.0%

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

7. Final simplification 26.0%

   \[\leadsto -m \]

Reproduce

herbie shell --seed 2023181 
(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))