a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 9.1s

Alternatives: 11

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, 0.8× speedup

Math

\[\begin{array}{l} \\ m \cdot \left(-1 + \left(\left(2 - m\right) + -1\right) \cdot \frac{m}{v}\right) \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (* m (+ -1.0 (* (+ (- 2.0 m) -1.0) (/ m v)))))

C

double code(double m, double v) {
	return m * (-1.0 + (((2.0 - 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) + (((2.0d0 - m) + (-1.0d0)) * (m / v)))
end function

Java

public static double code(double m, double v) {
	return m * (-1.0 + (((2.0 - m) + -1.0) * (m / v)));
}

Python

def code(m, v):
	return m * (-1.0 + (((2.0 - m) + -1.0) * (m / v)))

Julia

function code(m, v)
	return Float64(m * Float64(-1.0 + Float64(Float64(Float64(2.0 - m) + -1.0) * Float64(m / v))))
end

MATLAB

function tmp = code(m, v)
	tmp = m * (-1.0 + (((2.0 - m) + -1.0) * (m / v)));
end

Wolfram

code[m_, v_] := N[(m * N[(-1.0 + N[(N[(N[(2.0 - m), $MachinePrecision] + -1.0), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

4. Applied egg-rr 99.8%

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

   1. expm1-log1p-u 45.5%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

6. Applied egg-rr 45.5%

   \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]
7. Step-by-step derivation

   1. expm1-undefine 45.5%

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   2. sub-neg 45.5%

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   3. log1p-undefine 45.6%

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

   4. rem-exp-log 99.8%

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

   5. associate-+r- 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.2 \cdot 10^{-13}:\\ \;\;\;\;m \cdot \frac{m - v}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(\left(2 - m\right) + -1\right) \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.2e-13) (* m (/ (- m v) v)) (* m (* (+ (- 2.0 m) -1.0) (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.2e-13) {
		tmp = m * ((m - v) / v);
	} else {
		tmp = m * (((2.0 - 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 <= 4.2d-13) then
        tmp = m * ((m - v) / v)
    else
        tmp = m * (((2.0d0 - m) + (-1.0d0)) * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.2e-13) {
		tmp = m * ((m - v) / v);
	} else {
		tmp = m * (((2.0 - m) + -1.0) * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.2e-13:
		tmp = m * ((m - v) / v)
	else:
		tmp = m * (((2.0 - m) + -1.0) * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.2e-13)
		tmp = Float64(m * Float64(Float64(m - v) / v));
	else
		tmp = Float64(m * Float64(Float64(Float64(2.0 - m) + -1.0) * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.2e-13)
		tmp = m * ((m - v) / v);
	else
		tmp = m * (((2.0 - m) + -1.0) * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.2e-13], N[(m * N[(N[(m - v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(N[(2.0 - m), $MachinePrecision] + -1.0), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.19999999999999977e-13

   1. Initial program 99.7%

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

   3. Taylor expanded in m around inf 99.6%

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

   4. Taylor expanded in m around 0 99.4%

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

   5. Taylor expanded in v around 0 99.5%

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

      1. neg-mul-1 99.5%

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

      2. unsub-neg 99.5%

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

   7. Simplified 99.5%

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

   if 4.19999999999999977e-13 < 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. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fmm-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 4.1%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   6. Applied egg-rr 4.1%

      \[\leadsto m \cdot \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}, \frac{m}{v}, -1\right) \]
   7. Step-by-step derivation

      1. expm1-undefine 4.1%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      2. sub-neg 4.1%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      3. log1p-undefine 4.1%

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

      4. rem-exp-log 99.9%

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

      5. associate-+r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in v around 0 99.4%

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

       1. associate-/l* 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in m around 0 99.4%

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

       1. +-commutative 99.4%

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

       2. distribute-rgt-in 48.4%

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

       3. associate-/r/ 48.4%

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

       4. unpow-1 48.4%

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

       5. neg-mul-1 48.4%

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

       6. distribute-lft-neg-out 48.4%

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

       7. associate-/r/ 48.4%

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

       8. sub-neg 48.4%

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

       9. unpow-1 48.4%

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

       10. div-sub 99.4%

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

       11. associate-/r/ 99.4%

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

       12. *-commutative 99.4%

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

       13. associate-*r/ 99.4%

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

       14. associate-*l/ 99.4%

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

   14. Simplified 99.4%

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

       1. expm1-log1p-u 4.1%

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   16. Applied egg-rr 3.7%

       \[\leadsto m \cdot \left(\frac{m}{v} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}\right) \]
   17. Step-by-step derivation

       1. expm1-undefine 4.1%

          \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

       2. sub-neg 4.1%

          \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

       3. log1p-undefine 4.1%

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

       4. rem-exp-log 99.9%

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

       5. associate-+r- 99.9%

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

       6. metadata-eval 99.9%

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

       7. metadata-eval 99.9%

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

   18. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.2 \cdot 10^{-13}:\\ \;\;\;\;m \cdot \frac{m - v}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(\left(2 - m\right) + -1\right) \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.5 \cdot 10^{-177} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 4.5e-177) (not (<= m 1.0))) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 4.5e-177) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((m <= 4.5d-177) .or. (.not. (m <= 1.0d0))) then
        tmp = -m
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m <= 4.5e-177) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 4.5e-177) or not (m <= 1.0):
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((m <= 4.5e-177) || !(m <= 1.0))
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m <= 4.5e-177) || ~((m <= 1.0)))
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 4.5e-177], N[Not[LessEqual[m, 1.0]], $MachinePrecision]], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.5000000000000003e-177 or 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fmm-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 24.4%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   6. Applied egg-rr 24.4%

      \[\leadsto m \cdot \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}, \frac{m}{v}, -1\right) \]
   7. Step-by-step derivation

      1. expm1-undefine 24.4%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      2. sub-neg 24.4%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      3. log1p-undefine 24.4%

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

      4. rem-exp-log 99.9%

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

      5. associate-+r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in m around 0 25.1%

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

       1. neg-mul-1 25.1%

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

   11. Simplified 25.1%

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

   if 4.5000000000000003e-177 < 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. *-commutative 99.5%

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

      3. associate-/l* 99.5%

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

      4. fmm-def 99.5%

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

      5. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   6. Applied egg-rr 99.5%

      \[\leadsto m \cdot \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}, \frac{m}{v}, -1\right) \]
   7. Step-by-step derivation

      1. expm1-undefine 99.5%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      2. sub-neg 99.5%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      3. log1p-undefine 99.5%

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

      4. rem-exp-log 99.5%

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

      5. associate-+r- 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in v around 0 72.6%

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

       1. associate-/l* 72.5%

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

   11. Simplified 72.5%

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

   12. Taylor expanded in m around 0 69.2%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.5 \cdot 10^{-177} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4 \cdot 10^{-13}:\\ \;\;\;\;m \cdot \frac{m - v}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4e-13) (* m (/ (- m v) v)) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4e-13) {
		tmp = m * ((m - v) / v);
	} 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 <= 2.4d-13) then
        tmp = m * ((m - v) / v)
    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 <= 2.4e-13) {
		tmp = m * ((m - v) / v);
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4e-13:
		tmp = m * ((m - v) / v)
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4e-13)
		tmp = Float64(m * Float64(Float64(m - v) / v));
	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 <= 2.4e-13)
		tmp = m * ((m - v) / v);
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4e-13], N[(m * N[(N[(m - v), $MachinePrecision] / v), $MachinePrecision]), $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 < 2.3999999999999999e-13

   1. Initial program 99.7%

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

   3. Taylor expanded in m around inf 99.6%

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

   4. Taylor expanded in m around 0 99.4%

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

   5. Taylor expanded in v around 0 99.5%

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

      1. neg-mul-1 99.5%

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

      2. unsub-neg 99.5%

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

   7. Simplified 99.5%

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

   if 2.3999999999999999e-13 < 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. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fmm-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 4.1%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   6. Applied egg-rr 4.1%

      \[\leadsto m \cdot \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}, \frac{m}{v}, -1\right) \]
   7. Step-by-step derivation

      1. expm1-undefine 4.1%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      2. sub-neg 4.1%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      3. log1p-undefine 4.1%

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

      4. rem-exp-log 99.9%

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

      5. associate-+r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in v around 0 99.4%

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

       1. associate-/l* 99.4%

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

   11. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4 \cdot 10^{-13}:\\ \;\;\;\;m \cdot \frac{m - v}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in m around inf 99.6%

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

   4. Taylor expanded in m around 0 97.1%

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

   5. Taylor expanded in v around 0 97.2%

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

      1. neg-mul-1 97.2%

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

      2. unsub-neg 97.2%

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

   7. Simplified 97.2%

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

   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. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fmm-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   6. Applied egg-rr 0.0%

      \[\leadsto m \cdot \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}, \frac{m}{v}, -1\right) \]
   7. Step-by-step derivation

      1. expm1-undefine 0.0%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      2. sub-neg 0.0%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      3. log1p-undefine 0.0%

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

      4. rem-exp-log 99.9%

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

      5. associate-+r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in v around 0 99.9%

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

       1. associate-/l* 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in m around inf 98.0%

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

       1. neg-mul-1 98.0%

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

   14. Simplified 98.0%

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

4. Final simplification 97.6%

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

Alternative 6: 51.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in m around inf 99.6%

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

   4. Taylor expanded in m around 0 97.1%

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

   5. Taylor expanded in v around 0 97.2%

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

      1. neg-mul-1 97.2%

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

      2. unsub-neg 97.2%

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

   7. Simplified 97.2%

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

   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. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fmm-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   6. Applied egg-rr 0.0%

      \[\leadsto m \cdot \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}, \frac{m}{v}, -1\right) \]
   7. Step-by-step derivation

      1. expm1-undefine 0.0%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      2. sub-neg 0.0%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      3. log1p-undefine 0.0%

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

      4. rem-exp-log 99.9%

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

      5. associate-+r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in m around 0 5.8%

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

       1. neg-mul-1 5.8%

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

   11. Simplified 5.8%

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

4. Final simplification 47.6%

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

Alternative 7: 51.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0 97.1%

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

   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. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fmm-def 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   6. Applied egg-rr 0.0%

      \[\leadsto m \cdot \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}, \frac{m}{v}, -1\right) \]
   7. Step-by-step derivation

      1. expm1-undefine 0.0%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      2. sub-neg 0.0%

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

      3. log1p-undefine 0.0%

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

      4. rem-exp-log 99.9%

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

      5. associate-+r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in m around 0 5.8%

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

       1. neg-mul-1 5.8%

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

   11. Simplified 5.8%

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

4. Final simplification 47.6%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

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(m * Float64(-1.0 + Float64(Float64(m / v) * Float64(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[(m * N[(-1.0 + N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 10: 26.8% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-/l* 99.8%

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

   4. fmm-def 99.8%

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

   5. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 45.5%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

6. Applied egg-rr 45.5%

   \[\leadsto m \cdot \mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}, \frac{m}{v}, -1\right) \]
7. Step-by-step derivation

   1. expm1-undefine 45.5%

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   2. sub-neg 45.5%

      \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \cdot \frac{m}{v} - 1\right) \cdot m \]

   3. log1p-undefine 45.6%

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

   4. rem-exp-log 99.8%

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

   5. associate-+r- 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in m around 0 24.4%

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

    1. neg-mul-1 24.4%

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

11. Simplified 24.4%

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

Alternative 11: 3.0% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (m v) :precision binary64 m)

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return m

Julia

function code(m, v)
	return m
end

MATLAB

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

Wolfram

code[m_, v_] := m

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-/l* 99.8%

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

   4. fmm-def 99.8%

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

   5. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 24.4%

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

   1. *-commutative 24.4%

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

   2. neg-mul-1 24.4%

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

   3. neg-sub0 24.4%

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

   4. sub-neg 24.4%

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

   5. add-sqr-sqrt 0.0%

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

   6. sqrt-unprod 3.0%

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

   7. sqr-neg 3.0%

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

   8. sqrt-unprod 2.9%

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

   9. add-sqr-sqrt 2.9%

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

7. Applied egg-rr 2.9%

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

   1. +-lft-identity 2.9%

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

9. Simplified 2.9%

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

Reproduce

herbie shell --seed 2024163 
(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))