b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 12

Speedup: 1.0×

• 42.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
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. Final simplification 99.9%

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

Alternative 2: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m + \frac{m}{v}\\ \mathbf{if}\;m \leq 2.8 \cdot 10^{-147}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-99}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (+ m (/ m v))))
   (if (<= m 2.8e-147)
     -1.0
     (if (<= m 3.1e-120)
       t_0
       (if (<= m 5.2e-99) -1.0 (if (<= m 2.6) t_0 (* m (* m (/ m v)))))))))

C

double code(double m, double v) {
	double t_0 = m + (m / v);
	double tmp;
	if (m <= 2.8e-147) {
		tmp = -1.0;
	} else if (m <= 3.1e-120) {
		tmp = t_0;
	} else if (m <= 5.2e-99) {
		tmp = -1.0;
	} else if (m <= 2.6) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = m + (m / v)
    if (m <= 2.8d-147) then
        tmp = -1.0d0
    else if (m <= 3.1d-120) then
        tmp = t_0
    else if (m <= 5.2d-99) then
        tmp = -1.0d0
    else if (m <= 2.6d0) then
        tmp = t_0
    else
        tmp = m * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = m + (m / v);
	double tmp;
	if (m <= 2.8e-147) {
		tmp = -1.0;
	} else if (m <= 3.1e-120) {
		tmp = t_0;
	} else if (m <= 5.2e-99) {
		tmp = -1.0;
	} else if (m <= 2.6) {
		tmp = t_0;
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m + (m / v)
	tmp = 0
	if m <= 2.8e-147:
		tmp = -1.0
	elif m <= 3.1e-120:
		tmp = t_0
	elif m <= 5.2e-99:
		tmp = -1.0
	elif m <= 2.6:
		tmp = t_0
	else:
		tmp = m * (m * (m / v))
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m + Float64(m / v))
	tmp = 0.0
	if (m <= 2.8e-147)
		tmp = -1.0;
	elseif (m <= 3.1e-120)
		tmp = t_0;
	elseif (m <= 5.2e-99)
		tmp = -1.0;
	elseif (m <= 2.6)
		tmp = t_0;
	else
		tmp = Float64(m * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m + (m / v);
	tmp = 0.0;
	if (m <= 2.8e-147)
		tmp = -1.0;
	elseif (m <= 3.1e-120)
		tmp = t_0;
	elseif (m <= 5.2e-99)
		tmp = -1.0;
	elseif (m <= 2.6)
		tmp = t_0;
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 2.8e-147], -1.0, If[LessEqual[m, 3.1e-120], t$95$0, If[LessEqual[m, 5.2e-99], -1.0, If[LessEqual[m, 2.6], t$95$0, N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.8e-147 or 3.10000000000000019e-120 < m < 5.2000000000000001e-99

   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 79.4%

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

   if 2.8e-147 < m < 3.10000000000000019e-120 or 5.2000000000000001e-99 < 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 94.5%

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

      1. distribute-rgt-in 94.5%

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

      2. *-lft-identity 94.5%

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

      3. associate--l+ 94.5%

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

      4. associate-*l/ 94.8%

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

      5. *-lft-identity 94.8%

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

      6. sub-neg 94.8%

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

      7. metadata-eval 94.8%

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

   6. Simplified 94.8%

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

   7. Taylor expanded in m around inf 75.4%

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

      1. distribute-lft-in 75.4%

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

      2. /-rgt-identity 75.4%

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

      3. times-frac 75.6%

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

      4. *-commutative 75.6%

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

      5. times-frac 75.6%

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

      6. metadata-eval 75.6%

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

      7. *-rgt-identity 75.6%

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

      8. *-rgt-identity 75.6%

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

   9. Simplified 75.6%

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

   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 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in v around 0 99.3%

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

      1. unpow2 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r/ 99.3%

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

      4. associate-/l* 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. associate-/r/ 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-*l* 99.3%

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

      11. neg-sub0 99.3%

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

      12. associate--r- 99.3%

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

      13. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

   10. Taylor expanded in m around inf 99.3%

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

       1. unpow2 99.3%

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

       2. associate-/l* 99.3%

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

       3. associate-/r/ 99.3%

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

   12. Simplified 99.3%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{-147}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{-120}:\\ \;\;\;\;m + \frac{m}{v}\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-99}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;m + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * ((1.0 / v) * (m * (m + -1.0)));
	}
	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 = (1.0d0 - m) * ((-1.0d0) + (m / v))
    else
        tmp = m * ((1.0d0 / v) * (m * (m + (-1.0d0))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 97.6%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   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 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in v around 0 99.3%

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

      1. unpow2 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r/ 99.3%

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

      4. associate-/l* 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. associate-/r/ 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-*l* 99.3%

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

      11. neg-sub0 99.3%

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

      12. associate--r- 99.3%

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

      13. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

       1. associate-*l/ 99.3%

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

       2. clear-num 99.3%

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

       3. +-commutative 99.3%

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

   11. Applied egg-rr 99.3%

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

       1. associate-/r/ 99.3%

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

   13. Simplified 99.3%

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

4. Final simplification 98.5%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 97.6%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   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 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in v around 0 99.3%

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

      1. unpow2 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r/ 99.3%

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

      4. associate-/l* 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. associate-/r/ 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-*l* 99.3%

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

      11. neg-sub0 99.3%

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

      12. associate--r- 99.3%

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

      13. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

       1. associate-*l/ 99.3%

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

       2. clear-num 99.3%

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

       3. +-commutative 99.3%

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

   11. Applied egg-rr 99.3%

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

       1. +-commutative 99.3%

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

       2. distribute-rgt-in 99.3%

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

       3. remove-double-neg 99.3%

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

       4. neg-mul-1 99.3%

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

       5. unpow2 99.3%

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

       6. sub-neg 99.3%

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

       7. sub-neg 99.3%

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

       8. unpow2 99.3%

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

       9. neg-mul-1 99.3%

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

       10. remove-double-neg 99.3%

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

       11. distribute-rgt-in 99.3%

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

       12. +-commutative 99.3%

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

       13. distribute-rgt-in 99.3%

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

       14. neg-mul-1 99.3%

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

       15. unsub-neg 99.3%

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

   13. Simplified 99.3%

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

4. Final simplification 98.5%

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

Alternative 5: 61.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.5 \cdot 10^{-146}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-120} \lor \neg \left(m \leq 5.2 \cdot 10^{-99}\right):\\ \;\;\;\;m + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.5e-146)
   -1.0
   (if (or (<= m 2.3e-120) (not (<= m 5.2e-99))) (+ m (/ m v)) -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.5e-146) {
		tmp = -1.0;
	} else if ((m <= 2.3e-120) || !(m <= 5.2e-99)) {
		tmp = m + (m / v);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.5d-146) then
        tmp = -1.0d0
    else if ((m <= 2.3d-120) .or. (.not. (m <= 5.2d-99))) then
        tmp = m + (m / v)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.5e-146) {
		tmp = -1.0;
	} else if ((m <= 2.3e-120) || !(m <= 5.2e-99)) {
		tmp = m + (m / v);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.5e-146:
		tmp = -1.0
	elif (m <= 2.3e-120) or not (m <= 5.2e-99):
		tmp = m + (m / v)
	else:
		tmp = -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.5e-146)
		tmp = -1.0;
	elseif ((m <= 2.3e-120) || !(m <= 5.2e-99))
		tmp = Float64(m + Float64(m / v));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.5e-146)
		tmp = -1.0;
	elseif ((m <= 2.3e-120) || ~((m <= 5.2e-99)))
		tmp = m + (m / v);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.5e-146], -1.0, If[Or[LessEqual[m, 2.3e-120], N[Not[LessEqual[m, 5.2e-99]], $MachinePrecision]], N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if m < 2.49999999999999979e-146 or 2.29999999999999986e-120 < m < 5.2000000000000001e-99

   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 79.4%

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

   if 2.49999999999999979e-146 < m < 2.29999999999999986e-120 or 5.2000000000000001e-99 < 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 0 69.0%

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

      1. distribute-rgt-in 69.0%

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

      2. *-lft-identity 69.0%

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

      3. associate--l+ 69.0%

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

      4. associate-*l/ 69.1%

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

      5. *-lft-identity 69.1%

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

      6. sub-neg 69.1%

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

      7. metadata-eval 69.1%

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

   6. Simplified 69.1%

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

   7. Taylor expanded in m around inf 63.2%

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

      1. distribute-lft-in 63.2%

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

      2. /-rgt-identity 63.2%

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

      3. times-frac 63.3%

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

      4. *-commutative 63.3%

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

      5. times-frac 63.3%

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

      6. metadata-eval 63.3%

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

      7. *-rgt-identity 63.3%

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

      8. *-rgt-identity 63.3%

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

   9. Simplified 63.3%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.5 \cdot 10^{-146}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-120} \lor \neg \left(m \leq 5.2 \cdot 10^{-99}\right):\\ \;\;\;\;m + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 97.4%

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

      1. distribute-rgt-in 97.4%

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

      2. *-lft-identity 97.4%

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

      3. associate--l+ 97.4%

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

      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 1 < 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 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in v around 0 99.3%

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

      1. unpow2 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r/ 99.3%

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

      4. associate-/l* 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. associate-/r/ 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-*l* 99.3%

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

      11. neg-sub0 99.3%

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

      12. associate--r- 99.3%

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

      13. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 98.5%

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

Alternative 7: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = m + (-1.0 + (m / v))
	else:
		tmp = m * ((m + -1.0) / (v / 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 * Float64(Float64(m + -1.0) / Float64(v / 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 * ((m + -1.0) / (v / 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], N[(m * N[(N[(m + -1.0), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 97.4%

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

      1. distribute-rgt-in 97.4%

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

      2. *-lft-identity 97.4%

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

      3. associate--l+ 97.4%

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

      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 1 < 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 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in v around 0 99.3%

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

      1. unpow2 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r/ 99.3%

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

      4. associate-/l* 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. associate-/r/ 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-*l* 99.3%

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

      11. neg-sub0 99.3%

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

      12. associate--r- 99.3%

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

      13. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

       1. *-commutative 99.3%

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

       2. clear-num 99.3%

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

       3. un-div-inv 99.3%

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

       4. +-commutative 99.3%

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

   11. Applied egg-rr 99.3%

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

4. Final simplification 98.5%

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

Alternative 8: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 97.6%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   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 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in v around 0 99.3%

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

      1. unpow2 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r/ 99.3%

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

      4. associate-/l* 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. associate-/r/ 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-*l* 99.3%

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

      11. neg-sub0 99.3%

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

      12. associate--r- 99.3%

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

      13. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

       1. *-commutative 99.3%

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

       2. clear-num 99.3%

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

       3. un-div-inv 99.3%

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

       4. +-commutative 99.3%

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

   11. Applied egg-rr 99.3%

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

4. Final simplification 98.5%

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

Alternative 9: 97.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = m + (-1.0 + (m / 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 <= 2.6d0) then
        tmp = m + ((-1.0d0) + (m / 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 <= 2.6) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6], N[(m + N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $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 97.4%

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

      1. distribute-rgt-in 97.4%

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

      2. *-lft-identity 97.4%

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

      3. associate--l+ 97.4%

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

      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 2.60000000000000009 < 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 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in v around 0 99.3%

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

      1. unpow2 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r/ 99.3%

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

      4. associate-/l* 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. associate-/r/ 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-*l* 99.3%

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

      11. neg-sub0 99.3%

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

      12. associate--r- 99.3%

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

      13. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

   10. Taylor expanded in m around inf 99.3%

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

       1. unpow2 99.3%

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

       2. associate-/l* 99.3%

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

       3. associate-/r/ 99.3%

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

   12. Simplified 99.3%

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

4. Final simplification 98.4%

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

Alternative 10: 26.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.8 \cdot 10^{-51}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3.8e-51) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.8e-51) {
		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 <= 3.8d-51) 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 <= 3.8e-51) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.8e-51:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.8e-51)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.8e-51)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.8e-51], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 3.80000000000000003e-51

   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 59.6%

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

   if 3.80000000000000003e-51 < 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 0 13.4%

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

   5. Taylor expanded in m around inf 12.5%

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

      1. +-commutative 12.5%

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

      2. distribute-rgt-in 12.5%

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

      3. *-lft-identity 12.5%

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

      4. associate-+l+ 12.5%

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

      5. mul-1-neg 12.5%

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

      6. unpow2 12.5%

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

      7. associate-*l/ 12.5%

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

      8. distribute-lft-neg-in 12.5%

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

      9. distribute-rgt-in 12.5%

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

      10. sub-neg 12.5%

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

      11. div-sub 12.5%

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

   7. Simplified 12.5%

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

   8. Taylor expanded in v around inf 5.4%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.8 \cdot 10^{-51}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]

Alternative 11: 27.0% 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 25.8%

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

   1. neg-mul-1 25.8%

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

   2. neg-sub0 25.8%

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

   3. associate--r- 25.8%

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

   4. metadata-eval 25.8%

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

6. Simplified 25.8%

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

7. Final simplification 25.8%

   \[\leadsto m + -1 \]

Alternative 12: 24.6% 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 23.3%

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

5. Final simplification 23.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023275 
(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)))