b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.8%

Time: 5.0s

Alternatives: 13

Speedup: 1.0×

• 41.0% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.4 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{1 + m \cdot \left(m + -2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.4e-15)
   (* (- 1.0 m) (+ (/ m v) -1.0))
   (/ m (/ v (+ 1.0 (* m (+ m -2.0)))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.4e-15) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m / (v / (1.0 + (m * (m + -2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 3.4d-15) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = m / (v / (1.0d0 + (m * (m + (-2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.4e-15) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m / (v / (1.0 + (m * (m + -2.0))));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.4e-15:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = m / (v / (1.0 + (m * (m + -2.0))))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.4e-15)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m / Float64(v / Float64(1.0 + Float64(m * Float64(m + -2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.4e-15)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = m / (v / (1.0 + (m * (m + -2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.4e-15], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / N[(1.0 + N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.4e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 99.9%

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

   if 3.4e-15 < m

   1. Initial program 99.8%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in m around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. distribute-rgt-out 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.4 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{1 + m \cdot \left(m + -2\right)}}\\ \end{array} \]

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m - m \cdot m}{v} \cdot \left(1 - m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.1e-14)
   (* (- 1.0 m) (+ (/ m v) -1.0))
   (* (/ (- m (* m m)) v) (- 1.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.1e-14) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = ((m - (m * m)) / v) * (1.0 - 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 <= 1.1d-14) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = ((m - (m * m)) / v) * (1.0d0 - m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.1e-14) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = ((m - (m * m)) / v) * (1.0 - m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.1e-14:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = ((m - (m * m)) / v) * (1.0 - m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.1e-14)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(Float64(m - Float64(m * m)) / v) * Float64(1.0 - m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.1e-14)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = ((m - (m * m)) / v) * (1.0 - m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.1e-14], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.1e-14

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 99.9%

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

   if 1.1e-14 < m

   1. Initial program 99.8%

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

   2. Taylor expanded in m around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unpow2 99.9%

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

      3. distribute-lft-neg-out 99.9%

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

      4. +-commutative 99.9%

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

      5. cancel-sign-sub-inv 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. unpow2 99.9%

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

   7. Simplified 99.9%

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

      1. associate-/l* 99.9%

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

      2. div-inv 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. distribute-rgt-out-- 99.8%

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

      5. clear-num 99.8%

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

      6. distribute-rgt-out-- 99.9%

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

      7. *-un-lft-identity 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m - m \cdot m}{v} \cdot \left(1 - m\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{\left(1 - m\right) \cdot \left(1 - m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5.5e-15)
   (* (- 1.0 m) (+ (/ m v) -1.0))
   (/ m (/ v (* (- 1.0 m) (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.5e-15) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m / (v / ((1.0 - m) * (1.0 - 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 <= 5.5d-15) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = m / (v / ((1.0d0 - m) * (1.0d0 - m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5.5e-15) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m / (v / ((1.0 - m) * (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.5e-15:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = m / (v / ((1.0 - m) * (1.0 - m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.5e-15)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m / Float64(v / Float64(Float64(1.0 - m) * Float64(1.0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.5e-15)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = m / (v / ((1.0 - m) * (1.0 - m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.5e-15], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / N[(N[(1.0 - m), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.5000000000000002e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 99.9%

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

   if 5.5000000000000002e-15 < m

   1. Initial program 99.8%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

      1. unpow2 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{\left(1 - m\right) \cdot \left(1 - m\right)}}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 - m), $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. Taylor expanded in m around 0 99.9%

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

   1. mul-1-neg 99.9%

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

   2. unpow2 99.9%

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

   3. distribute-lft-neg-out 99.9%

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

   4. +-commutative 99.9%

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

   5. cancel-sign-sub-inv 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m * N[(N[(1.0 - m), $MachinePrecision] / 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. Taylor expanded in m around 0 71.8%

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

   1. mul-1-neg 71.8%

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

   2. unsub-neg 71.8%

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

   3. div-sub 99.9%

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

   4. *-rgt-identity 99.9%

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

   5. unpow2 99.9%

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

   6. distribute-lft-out-- 99.9%

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

   7. *-commutative 99.9%

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

   8. associate-*l/ 99.8%

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

   9. *-commutative 99.8%

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

4. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 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. Taylor expanded in m around 0 71.8%

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

   1. mul-1-neg 71.8%

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

   2. unsub-neg 71.8%

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

   3. div-sub 99.9%

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

   4. *-rgt-identity 99.9%

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

   5. unpow2 99.9%

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

   6. distribute-lft-out-- 99.9%

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

   7. *-commutative 99.9%

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

   8. associate-*l/ 99.8%

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

   9. *-commutative 99.8%

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

4. Simplified 99.8%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 7: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.42:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.42) (* (- 1.0 m) (+ (/ m v) -1.0)) (/ m (/ v (* m m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.42) {
		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 <= 0.42d0) 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 <= 0.42) {
		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 <= 0.42:
		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 <= 0.42)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m / Float64(v / Float64(m * m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.42)
		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, 0.42], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.419999999999999984

   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 96.6%

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

   if 0.419999999999999984 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in m around inf 95.4%

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

      1. unpow2 95.4%

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

   7. Simplified 95.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.42:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot m}}\\ \end{array} \]

Alternative 8: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.65:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(m + -2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.65) (* (- 1.0 m) (+ (/ m v) -1.0)) (/ m (/ v (* m (+ m -2.0))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.65) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m / (v / (m * (m + -2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.65d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = m / (v / (m * (m + (-2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.65) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m / (v / (m * (m + -2.0)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.65:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = m / (v / (m * (m + -2.0)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.65)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m / Float64(v / Float64(m * Float64(m + -2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.65)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = m / (v / (m * (m + -2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.65], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6499999999999999

   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 96.0%

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

   if 1.6499999999999999 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in m around inf 97.1%

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

      1. +-commutative 97.1%

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

      2. unpow2 97.1%

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

      3. distribute-rgt-out 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.65:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(m + -2\right)}}\\ \end{array} \]

Alternative 9: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.38) (+ (/ m v) (+ m -1.0)) (/ m (/ v (* m m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = (m / v) + (m + -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 <= 0.38d0) then
        tmp = (m / v) + (m + (-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 <= 0.38) {
		tmp = (m / v) + (m + -1.0);
	} else {
		tmp = m / (v / (m * m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.38:
		tmp = (m / v) + (m + -1.0)
	else:
		tmp = m / (v / (m * m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.38)
		tmp = Float64(Float64(m / v) + Float64(m + -1.0));
	else
		tmp = Float64(m / Float64(v / Float64(m * m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.38)
		tmp = (m / v) + (m + -1.0);
	else
		tmp = m / (v / (m * m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.38], N[(N[(m / v), $MachinePrecision] + N[(m + -1.0), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.38

   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 96.3%

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

      1. sub-neg 96.3%

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

      2. metadata-eval 96.3%

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

      3. +-commutative 96.3%

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

      4. +-commutative 96.3%

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

      5. distribute-rgt1-in 96.3%

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

      6. associate-*l/ 96.5%

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

      7. *-lft-identity 96.5%

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

      8. associate-+r+ 96.5%

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

   4. Simplified 96.5%

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

   if 0.38 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in m around inf 95.4%

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

      1. unpow2 95.4%

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

   7. Simplified 95.4%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot m}}\\ \end{array} \]

Alternative 10: 76.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(m / v), $MachinePrecision] + N[(m + -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. Taylor expanded in m around 0 77.4%

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

   1. sub-neg 77.4%

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

   2. metadata-eval 77.4%

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

   3. +-commutative 77.4%

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

   4. +-commutative 77.4%

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

   5. distribute-rgt1-in 77.4%

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

   6. associate-*l/ 77.5%

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

   7. *-lft-identity 77.5%

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

   8. associate-+r+ 77.5%

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

4. Simplified 77.5%

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

5. Final simplification 77.5%

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

Alternative 11: 60.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3 \cdot 10^{-196}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3e-196) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3e-196) {
		tmp = -1.0;
	} else {
		tmp = 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 <= 3d-196) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3e-196) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3e-196:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3e-196)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3e-196)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3e-196], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3e-196

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 81.7%

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

   if 3e-196 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in v around 0 89.3%

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

      1. associate-/l* 89.3%

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

   4. Simplified 89.3%

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

   5. Taylor expanded in m around 0 89.3%

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

      1. +-commutative 89.3%

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

      2. unpow2 89.3%

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

      3. distribute-rgt-out 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in m around 0 60.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3 \cdot 10^{-196}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 12: 28.1% 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. Taylor expanded in v around inf 28.1%

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

   1. mul-1-neg 28.1%

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

   2. sub-neg 28.1%

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

   3. distribute-neg-in 28.1%

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

   4. metadata-eval 28.1%

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

   5. remove-double-neg 28.1%

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

4. Simplified 28.1%

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

5. Final simplification 28.1%

   \[\leadsto m + -1 \]

Alternative 13: 25.7% 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. Taylor expanded in m around 0 25.7%

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

3. Final simplification 25.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023187 
(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)))