b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

Alternatives: 11

Speedup: 1.0×

• 41.5% 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

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

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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / N[((-v) / 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. Taylor expanded in m around 0 97.3%

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

   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 99.2%

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

      1. associate-*r/ 99.2%

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

      2. neg-mul-1 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 98.2%

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

Alternative 3: 83.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-187}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;m + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.9e-187) -1.0 (if (<= m 2.6) (+ m (/ m v)) (* m (* m (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-187) {
		tmp = -1.0;
	} else if (m <= 2.6) {
		tmp = m + (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 <= 2.9d-187) then
        tmp = -1.0d0
    else if (m <= 2.6d0) then
        tmp = m + (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 <= 2.9e-187) {
		tmp = -1.0;
	} else if (m <= 2.6) {
		tmp = m + (m / v);
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.9e-187:
		tmp = -1.0
	elif m <= 2.6:
		tmp = m + (m / v)
	else:
		tmp = m * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.9e-187)
		tmp = -1.0;
	elseif (m <= 2.6)
		tmp = Float64(m + 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 <= 2.9e-187)
		tmp = -1.0;
	elseif (m <= 2.6)
		tmp = m + (m / v);
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.9e-187], -1.0, If[LessEqual[m, 2.6], N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.89999999999999988e-187

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 83.6%

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

   if 2.89999999999999988e-187 < m < 2.60000000000000009

   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%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 94.3%

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

   5. Taylor expanded in m around inf 77.6%

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

   6. Taylor expanded in v around 0 77.8%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.9%

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

      1. cube-mult 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. times-frac 99.8%

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

      4. /-rgt-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-187}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;m + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 4: 83.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.5 \cdot 10^{-188}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7.5e-188)
   -1.0
   (if (<= m 1.0) (* (- 1.0 m) (/ m v)) (* m (* m (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.5e-188) {
		tmp = -1.0;
	} else if (m <= 1.0) {
		tmp = (1.0 - m) * (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 <= 7.5d-188) then
        tmp = -1.0d0
    else if (m <= 1.0d0) then
        tmp = (1.0d0 - m) * (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 <= 7.5e-188) {
		tmp = -1.0;
	} else if (m <= 1.0) {
		tmp = (1.0 - m) * (m / v);
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.5e-188:
		tmp = -1.0
	elif m <= 1.0:
		tmp = (1.0 - m) * (m / v)
	else:
		tmp = m * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.5e-188)
		tmp = -1.0;
	elseif (m <= 1.0)
		tmp = Float64(Float64(1.0 - m) * 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 <= 7.5e-188)
		tmp = -1.0;
	elseif (m <= 1.0)
		tmp = (1.0 - m) * (m / v);
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.5e-188], -1.0, If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 7.5000000000000001e-188

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 83.6%

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

   if 7.5000000000000001e-188 < m < 1

   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 95.6%

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

   3. Taylor expanded in m around inf 78.6%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.2%

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

      1. cube-mult 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. times-frac 99.1%

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

      4. /-rgt-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. associate-/l* 99.2%

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

      2. associate-/r/ 99.2%

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

   8. Applied egg-rr 99.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.5 \cdot 10^{-188}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m / N[(N[(v / m), $MachinePrecision] / m), $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. Taylor expanded in m around 0 97.3%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.2%

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

      1. cube-mult 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. times-frac 99.1%

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

      4. /-rgt-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. associate-/l* 99.2%

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

      2. associate-/r/ 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r* 99.1%

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

      3. associate-/r/ 99.1%

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

      4. associate-/r* 99.2%

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

   10. Applied egg-rr 99.2%

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

4. Final simplification 98.2%

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

Alternative 6: 97.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6)
		tmp = Float64(-1.0 + Float64(m + 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 = -1.0 + (m + (m / v));
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6], N[(-1.0 + N[(m + 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 96.5%

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

      1. sub-neg 96.5%

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

      2. *-commutative 96.5%

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

      3. distribute-rgt-in 96.5%

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

      4. *-lft-identity 96.5%

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

      5. remove-double-neg 96.5%

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

      6. distribute-rgt-neg-out 96.5%

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

      7. *-commutative 96.5%

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

      8. associate-*r/ 96.7%

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

      9. *-rgt-identity 96.7%

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

      10. distribute-frac-neg 96.7%

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

      11. remove-double-neg 96.7%

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

      12. metadata-eval 96.7%

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

   6. Simplified 96.7%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.9%

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

      1. cube-mult 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. times-frac 99.8%

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

      4. /-rgt-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 98.1%

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

Alternative 7: 97.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 96.5%

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

      1. sub-neg 96.5%

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

      2. *-commutative 96.5%

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

      3. distribute-rgt-in 96.5%

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

      4. *-lft-identity 96.5%

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

      5. remove-double-neg 96.5%

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

      6. distribute-rgt-neg-out 96.5%

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

      7. *-commutative 96.5%

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

      8. associate-*r/ 96.7%

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

      9. *-rgt-identity 96.7%

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

      10. distribute-frac-neg 96.7%

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

      11. remove-double-neg 96.7%

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

      12. metadata-eval 96.7%

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

   6. Simplified 96.7%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.9%

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

      1. cube-mult 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. times-frac 99.8%

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

      4. /-rgt-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.8%

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

      3. associate-/r/ 99.9%

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

      4. associate-/r* 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 98.1%

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

Alternative 8: 60.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-187}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.5e-187) -1.0 (+ m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.5e-187) {
		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 <= 1.5d-187) 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 <= 1.5e-187) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.5e-187:
		tmp = -1.0
	else:
		tmp = m + (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.5e-187)
		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 <= 1.5e-187)
		tmp = -1.0;
	else
		tmp = m + (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.5e-187], -1.0, N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.50000000000000002e-187

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 83.6%

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

   if 1.50000000000000002e-187 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 75.1%

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

   5. Taylor expanded in m around inf 67.9%

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

   6. Taylor expanded in v around 0 68.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-187}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \]

Alternative 9: 26.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.26 \cdot 10^{-75}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.26e-75) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.26e-75) {
		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 <= 1.26d-75) 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 <= 1.26e-75) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.26e-75:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.26e-75)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.26e-75)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.26e-75], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 1.26000000000000002e-75

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.9%

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

   if 1.26000000000000002e-75 < 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/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 67.6%

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

   5. Taylor expanded in m around inf 66.3%

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

   6. Taylor expanded in v around inf 5.6%

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

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.26 \cdot 10^{-75}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]

Alternative 10: 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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around inf 25.5%

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

   1. neg-mul-1 25.5%

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

   2. neg-sub0 25.5%

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

   3. associate--r- 25.5%

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

   4. metadata-eval 25.5%

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

6. Simplified 25.5%

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

7. Final simplification 25.5%

   \[\leadsto m + -1 \]

Alternative 11: 24.5% 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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 23.1%

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

5. Final simplification 23.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023249 
(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)))