b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.0s

Alternatives: 14

Speedup: 1.0×

• 41.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\frac{m}{\frac{v}{1 - m}} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \frac{m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6)
   (+ (/ m (/ v (- 1.0 m))) (+ m -1.0))
   (* (+ m -2.0) (/ (* m m) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (m / (v / (1.0 - m))) + (m + -1.0);
	} else {
		tmp = (m + -2.0) * ((m * m) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (m / (v / (1.0d0 - m))) + (m + (-1.0d0))
    else
        tmp = (m + (-2.0d0)) * ((m * m) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (m / (v / (1.0 - m))) + (m + -1.0);
	} else {
		tmp = (m + -2.0) * ((m * m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (m / (v / (1.0 - m))) + (m + -1.0)
	else:
		tmp = (m + -2.0) * ((m * m) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(m / Float64(v / Float64(1.0 - m))) + Float64(m + -1.0));
	else
		tmp = Float64(Float64(m + -2.0) * Float64(Float64(m * m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (m / (v / (1.0 - m))) + (m + -1.0);
	else
		tmp = (m + -2.0) * ((m * m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(m + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m + -2.0), $MachinePrecision] * N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 97.6%

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

   3. Taylor expanded in v around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. mul-1-neg 97.6%

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

      3. unsub-neg 97.6%

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

      4. associate-/l* 97.6%

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

   5. Simplified 97.6%

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

   if 1.6000000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 23.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   7. Step-by-step derivation

      1. unpow2 23.1%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. associate-*r/ 23.1%

         \[\leadsto -2 \cdot \color{blue}{\left(m \cdot \frac{m}{v}\right)} + \frac{{m}^{3}}{v} \]

      3. cube-mult 23.0%

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

      4. associate-*r/ 23.1%

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

      5. associate-*r/ 23.1%

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

      6. distribute-rgt-out 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in m around 0 23.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   10. Step-by-step derivation

       1. *-commutative 23.1%

          \[\leadsto \color{blue}{\frac{{m}^{2}}{v} \cdot -2} + \frac{{m}^{3}}{v} \]

       2. unpow2 23.1%

          \[\leadsto \frac{\color{blue}{m \cdot m}}{v} \cdot -2 + \frac{{m}^{3}}{v} \]

       3. associate-*r/ 23.1%

          \[\leadsto \color{blue}{\left(m \cdot \frac{m}{v}\right)} \cdot -2 + \frac{{m}^{3}}{v} \]

       4. cube-mult 23.0%

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

       5. associate-*l/ 23.0%

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

       6. associate-*r* 23.1%

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

       7. *-commutative 23.1%

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

       8. distribute-lft-in 99.2%

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

       9. +-commutative 99.2%

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

       10. associate-*r/ 99.2%

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

   11. Simplified 99.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\frac{m}{\frac{v}{1 - m}} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \frac{m \cdot m}{v}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-27}:\\ \;\;\;\;m + \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m + m \cdot \left(m \cdot \left(m + -2\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-27) (+ m (+ (/ m v) -1.0)) (/ (+ m (* m (* m (+ m -2.0)))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-27) {
		tmp = m + ((m / v) + -1.0);
	} else {
		tmp = (m + (m * (m * (m + -2.0)))) / 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 <= 5d-27) then
        tmp = m + ((m / v) + (-1.0d0))
    else
        tmp = (m + (m * (m * (m + (-2.0d0))))) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5e-27) {
		tmp = m + ((m / v) + -1.0);
	} else {
		tmp = (m + (m * (m * (m + -2.0)))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5e-27:
		tmp = m + ((m / v) + -1.0)
	else:
		tmp = (m + (m * (m * (m + -2.0)))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-27)
		tmp = Float64(m + Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m + Float64(m * Float64(m * Float64(m + -2.0)))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-27)
		tmp = m + ((m / v) + -1.0);
	else
		tmp = (m + (m * (m * (m + -2.0)))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-27], N[(m + N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m + N[(m * N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.0000000000000002e-27

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 99.0%

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

      1. distribute-rgt-in 99.0%

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

      2. *-lft-identity 99.0%

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

      3. associate--l+ 99.0%

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

      4. associate-*l/ 100.0%

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

      5. *-lft-identity 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if 5.0000000000000002e-27 < m

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in m around inf 36.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \left(m \cdot \left(1 + \frac{1}{v}\right) + \frac{{m}^{3}}{v}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 36.1%

         \[\leadsto -2 \cdot \frac{{m}^{2}}{v} + \color{blue}{\left(\frac{{m}^{3}}{v} + m \cdot \left(1 + \frac{1}{v}\right)\right)} \]

      2. associate-+r+ 36.2%

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}\right) + m \cdot \left(1 + \frac{1}{v}\right)} \]

      3. unpow2 36.2%

         \[\leadsto \left(-2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v}\right) + m \cdot \left(1 + \frac{1}{v}\right) \]

      4. associate-*r/ 36.2%

         \[\leadsto \left(-2 \cdot \color{blue}{\left(m \cdot \frac{m}{v}\right)} + \frac{{m}^{3}}{v}\right) + m \cdot \left(1 + \frac{1}{v}\right) \]

      5. cube-mult 36.1%

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

      6. associate-*r/ 36.2%

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

      7. associate-*r/ 36.2%

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

      8. distribute-rgt-out 99.8%

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

      9. distribute-lft-in 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. associate-*r/ 99.8%

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

      12. *-rgt-identity 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in v around 0 99.8%

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

       1. unpow2 99.8%

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

       2. sub-neg 99.8%

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

       3. metadata-eval 99.8%

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

       4. associate-*l* 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-27}:\\ \;\;\;\;m + \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m + m \cdot \left(m \cdot \left(m + -2\right)\right)}{v}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. associate-/r/ 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 5: 83.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-190}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;m + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6e-190) -1.0 (if (<= m 2.6) (+ m (/ m v)) (* (/ m v) (* m m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6e-190) {
		tmp = -1.0;
	} else if (m <= 2.6) {
		tmp = m + (m / v);
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 6d-190) then
        tmp = -1.0d0
    else if (m <= 2.6d0) then
        tmp = m + (m / v)
    else
        tmp = (m / v) * (m * m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 6e-190) {
		tmp = -1.0;
	} else if (m <= 2.6) {
		tmp = m + (m / v);
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6e-190:
		tmp = -1.0
	elif m <= 2.6:
		tmp = m + (m / v)
	else:
		tmp = (m / v) * (m * m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6e-190)
		tmp = -1.0;
	elseif (m <= 2.6)
		tmp = Float64(m + Float64(m / v));
	else
		tmp = Float64(Float64(m / v) * Float64(m * m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6e-190)
		tmp = -1.0;
	elseif (m <= 2.6)
		tmp = m + (m / v);
	else
		tmp = (m / v) * (m * m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6e-190], -1.0, If[LessEqual[m, 2.6], N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 5.9999999999999996e-190

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 86.3%

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

   if 5.9999999999999996e-190 < m < 2.60000000000000009

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 94.9%

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

   5. Taylor expanded in m around inf 74.1%

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

      1. +-commutative 74.1%

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

      2. distribute-lft-in 74.1%

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

      3. div-inv 74.4%

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

      4. *-rgt-identity 74.4%

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

   7. Applied egg-rr 74.4%

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

   if 2.60000000000000009 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.7%

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   5. Step-by-step derivation

      1. div-inv 99.6%

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{v}} \]

      2. unpow3 99.6%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-190}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;m + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]

Alternative 6: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;m + \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (+ m (+ (/ m v) -1.0)) (* (+ m -2.0) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = m + ((m / v) + -1.0);
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.4d0) then
        tmp = m + ((m / v) + (-1.0d0))
    else
        tmp = (m + (-2.0d0)) * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = m + ((m / v) + -1.0);
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = m + ((m / v) + -1.0)
	else:
		tmp = (m + -2.0) * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(m + Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m + -2.0) * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = m + ((m / v) + -1.0);
	else
		tmp = (m + -2.0) * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(m + N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m + -2.0), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 96.6%

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

      1. distribute-rgt-in 96.6%

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

      2. *-lft-identity 96.6%

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

      3. associate--l+ 96.6%

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

      4. associate-*l/ 97.5%

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

      5. *-lft-identity 97.5%

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

      6. sub-neg 97.5%

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

      7. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

   if 2.39999999999999991 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 23.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   7. Step-by-step derivation

      1. unpow2 23.1%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. associate-*r/ 23.1%

         \[\leadsto -2 \cdot \color{blue}{\left(m \cdot \frac{m}{v}\right)} + \frac{{m}^{3}}{v} \]

      3. cube-mult 23.0%

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

      4. associate-*r/ 23.1%

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

      5. associate-*r/ 23.1%

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

      6. distribute-rgt-out 99.2%

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

   8. Simplified 99.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;m + \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- 1.0 m) (+ (/ m v) -1.0)) (* (+ m -2.0) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = (m + (-2.0d0)) * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = (m + -2.0) * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m + -2.0) * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = (m + -2.0) * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m + -2.0), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 97.6%

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

   if 1.6000000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 23.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   7. Step-by-step derivation

      1. unpow2 23.1%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. associate-*r/ 23.1%

         \[\leadsto -2 \cdot \color{blue}{\left(m \cdot \frac{m}{v}\right)} + \frac{{m}^{3}}{v} \]

      3. cube-mult 23.0%

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

      4. associate-*r/ 23.1%

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

      5. associate-*r/ 23.1%

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

      6. distribute-rgt-out 99.2%

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

   8. Simplified 99.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 8: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \frac{m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- 1.0 m) (+ (/ m v) -1.0)) (* (+ m -2.0) (/ (* m m) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m + -2.0) * ((m * m) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = (m + (-2.0d0)) * ((m * m) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m + -2.0) * ((m * m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = (m + -2.0) * ((m * m) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m + -2.0) * Float64(Float64(m * m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = (m + -2.0) * ((m * m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m + -2.0), $MachinePrecision] * N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 97.6%

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

   if 1.6000000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 23.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   7. Step-by-step derivation

      1. unpow2 23.1%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. associate-*r/ 23.1%

         \[\leadsto -2 \cdot \color{blue}{\left(m \cdot \frac{m}{v}\right)} + \frac{{m}^{3}}{v} \]

      3. cube-mult 23.0%

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

      4. associate-*r/ 23.1%

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

      5. associate-*r/ 23.1%

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

      6. distribute-rgt-out 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in m around 0 23.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   10. Step-by-step derivation

       1. *-commutative 23.1%

          \[\leadsto \color{blue}{\frac{{m}^{2}}{v} \cdot -2} + \frac{{m}^{3}}{v} \]

       2. unpow2 23.1%

          \[\leadsto \frac{\color{blue}{m \cdot m}}{v} \cdot -2 + \frac{{m}^{3}}{v} \]

       3. associate-*r/ 23.1%

          \[\leadsto \color{blue}{\left(m \cdot \frac{m}{v}\right)} \cdot -2 + \frac{{m}^{3}}{v} \]

       4. cube-mult 23.0%

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

       5. associate-*l/ 23.0%

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

       6. associate-*r* 23.1%

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

       7. *-commutative 23.1%

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

       8. distribute-lft-in 99.2%

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

       9. +-commutative 99.2%

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

       10. associate-*r/ 99.2%

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

   11. Simplified 99.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \frac{m \cdot m}{v}\\ \end{array} \]

Alternative 9: 73.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.15 \cdot 10^{-189}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;m + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.15e-189) -1.0 (if (<= m 0.27) (+ m (/ m v)) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.15e-189) {
		tmp = -1.0;
	} else if (m <= 0.27) {
		tmp = m + (m / v);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.15d-189) then
        tmp = -1.0d0
    else if (m <= 0.27d0) then
        tmp = m + (m / v)
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.15e-189) {
		tmp = -1.0;
	} else if (m <= 0.27) {
		tmp = m + (m / v);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.15e-189:
		tmp = -1.0
	elif m <= 0.27:
		tmp = m + (m / v)
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.15e-189)
		tmp = -1.0;
	elseif (m <= 0.27)
		tmp = Float64(m + Float64(m / v));
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.15e-189)
		tmp = -1.0;
	elseif (m <= 0.27)
		tmp = m + (m / v);
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.15e-189], -1.0, If[LessEqual[m, 0.27], N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.1499999999999999e-189

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 86.3%

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

   if 1.1499999999999999e-189 < m < 0.27000000000000002

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 95.8%

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

   5. Taylor expanded in m around inf 74.7%

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

      1. +-commutative 74.7%

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

      2. distribute-lft-in 74.7%

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

      3. div-inv 75.0%

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

      4. *-rgt-identity 75.0%

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

   7. Applied egg-rr 75.0%

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

   if 0.27000000000000002 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 0.1%

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

      1. sub-neg 0.1%

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

      2. distribute-lft-in 0.1%

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

      3. *-commutative 0.1%

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

      4. *-un-lft-identity 0.1%

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

      5. sub-neg 0.1%

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

      6. metadata-eval 0.1%

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

      7. +-commutative 0.1%

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

      8. sub-neg 0.1%

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

      9. metadata-eval 0.1%

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

      10. +-commutative 0.1%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 77.5%

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

      13. sqr-neg 77.5%

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

      14. sqrt-unprod 77.5%

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

      15. add-sqr-sqrt 77.5%

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

   4. Applied egg-rr 77.5%

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

      1. *-commutative 77.5%

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

      2. distribute-rgt1-in 77.5%

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

      3. +-commutative 77.5%

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

   6. Simplified 77.5%

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

   7. Taylor expanded in m around inf 77.5%

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

      1. unpow2 77.5%

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

      2. associate-*r/ 77.5%

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

   9. Simplified 77.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.15 \cdot 10^{-189}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;m + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 10: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;m + \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6) (+ m (+ (/ m v) -1.0)) (* (/ m v) (* m m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = m + ((m / v) + -1.0);
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.6d0) then
        tmp = m + ((m / v) + (-1.0d0))
    else
        tmp = (m / v) * (m * m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = m + ((m / v) + -1.0);
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6:
		tmp = m + ((m / v) + -1.0)
	else:
		tmp = (m / v) * (m * m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6)
		tmp = Float64(m + Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m / v) * Float64(m * m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6)
		tmp = m + ((m / v) + -1.0);
	else
		tmp = (m / v) * (m * m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6], N[(m + N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000009

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 96.0%

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

      1. distribute-rgt-in 96.0%

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

      2. *-lft-identity 96.0%

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

      3. associate--l+ 96.0%

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

      4. associate-*l/ 96.9%

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

      5. *-lft-identity 96.9%

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

      6. sub-neg 96.9%

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

      7. metadata-eval 96.9%

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

   6. Simplified 96.9%

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

   if 2.60000000000000009 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.7%

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   5. Step-by-step derivation

      1. div-inv 99.6%

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{v}} \]

      2. unpow3 99.6%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;m + \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]

Alternative 11: 62.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.65e-28) -1.0 (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-28) {
		tmp = -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 <= 2.65d-28) then
        tmp = -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 <= 2.65e-28) {
		tmp = -1.0;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.65e-28:
		tmp = -1.0
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.65e-28)
		tmp = -1.0;
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.65e-28)
		tmp = -1.0;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.65e-28], -1.0, N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.64999999999999994e-28

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 51.7%

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

   if 2.64999999999999994e-28 < m

   1. Initial program 99.8%

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

   2. Taylor expanded in m around 0 13.8%

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

      1. sub-neg 13.8%

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

      2. distribute-lft-in 13.8%

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

      3. *-commutative 13.8%

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

      4. *-un-lft-identity 13.8%

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

      5. sub-neg 13.8%

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

      6. metadata-eval 13.8%

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

      7. +-commutative 13.8%

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

      8. sub-neg 13.8%

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

      9. metadata-eval 13.8%

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

      10. +-commutative 13.8%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 78.4%

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

      13. sqr-neg 78.4%

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

      14. sqrt-unprod 78.4%

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

      15. add-sqr-sqrt 78.4%

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

   4. Applied egg-rr 78.4%

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

      1. *-commutative 78.4%

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

      2. distribute-rgt1-in 78.4%

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

      3. +-commutative 78.4%

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

   6. Simplified 78.4%

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

   7. Taylor expanded in m around inf 66.5%

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

      1. unpow2 66.5%

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

      2. associate-*r/ 66.5%

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

   9. Simplified 66.5%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 12: 27.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.02 \cdot 10^{-26}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.02e-26) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.02e-26) {
		tmp = -1.0;
	} 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 <= 1.02d-26) then
        tmp = -1.0d0
    else
        tmp = m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.02e-26) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.02e-26:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.02e-26)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.02e-26)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.02e-26], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 1.02e-26

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 51.7%

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

   if 1.02e-26 < m

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 56.9%

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

   5. Taylor expanded in m around inf 56.9%

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

   6. Taylor expanded in v around inf 5.7%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.02 \cdot 10^{-26}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]

Alternative 13: 27.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ m + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around inf 27.1%

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

   1. neg-mul-1 27.1%

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

   2. neg-sub0 27.1%

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

   3. associate--r- 27.1%

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

   4. metadata-eval 27.1%

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

6. Simplified 27.1%

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

7. Final simplification 27.1%

   \[\leadsto m + -1 \]

Alternative 14: 24.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 24.8%

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

5. Final simplification 24.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023290 
(FPCore (m v)
  :name "b 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) (- 1.0 m)))