b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.1%

Time: 8.0s

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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9 \cdot 10^{-45}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 9e-45) (+ -1.0 (+ m (/ m v))) (/ (* m (- 1.0 m)) (/ v (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 9e-45) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (1.0 - 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 <= 9d-45) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m * (1.0d0 - m)) / (v / (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 9e-45) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (1.0 - m)) / (v / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 9e-45:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m * (1.0 - m)) / (v / (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 9e-45)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(1.0 - m)) / Float64(v / Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 9e-45)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m * (1.0 - m)) / (v / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 9e-45], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.9999999999999997e-45

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. distribute-lft-in 99.8%

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

      3. div-inv 100.0%

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

      4. *-rgt-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 8.9999999999999997e-45 < 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. Add Preprocessing

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-*l/ 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*l/ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

      4. /-rgt-identity 99.9%

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

      5. associate-/r/ 99.9%

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

      6. associate-*l/ 99.9%

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

      7. associate-/l/ 99.9%

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

      8. associate-*l/ 99.9%

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

      9. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9 \cdot 10^{-45}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{1 - m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.8%

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

   2. associate-/r/ 99.8%

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

   3. clear-num 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 5: 60.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-164}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 4.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{-103}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.1e-164)
   -1.0
   (if (<= m 4.5e-129) (/ m v) (if (<= m 3.1e-103) -1.0 (+ m (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-164) {
		tmp = -1.0;
	} else if (m <= 4.5e-129) {
		tmp = m / v;
	} else if (m <= 3.1e-103) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.1d-164) then
        tmp = -1.0d0
    else if (m <= 4.5d-129) then
        tmp = m / v
    else if (m <= 3.1d-103) then
        tmp = -1.0d0
    else
        tmp = m + (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-164) {
		tmp = -1.0;
	} else if (m <= 4.5e-129) {
		tmp = m / v;
	} else if (m <= 3.1e-103) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.1e-164:
		tmp = -1.0
	elif m <= 4.5e-129:
		tmp = m / v
	elif m <= 3.1e-103:
		tmp = -1.0
	else:
		tmp = m + (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.1e-164)
		tmp = -1.0;
	elseif (m <= 4.5e-129)
		tmp = Float64(m / v);
	elseif (m <= 3.1e-103)
		tmp = -1.0;
	else
		tmp = Float64(m + Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.1e-164)
		tmp = -1.0;
	elseif (m <= 4.5e-129)
		tmp = m / v;
	elseif (m <= 3.1e-103)
		tmp = -1.0;
	else
		tmp = m + (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.1e-164], -1.0, If[LessEqual[m, 4.5e-129], N[(m / v), $MachinePrecision], If[LessEqual[m, 3.1e-103], -1.0, N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.0999999999999999e-164 or 4.50000000000000031e-129 < m < 3.1000000000000001e-103

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 86.8%

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

   if 2.0999999999999999e-164 < m < 4.50000000000000031e-129

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in v around 0 70.5%

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

      1. associate-*l/ 70.5%

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

   7. Simplified 70.5%

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

      1. associate-*l/ 70.5%

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

      2. unpow2 70.5%

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

      3. associate-*r* 70.5%

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

      4. /-rgt-identity 70.5%

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

      5. associate-/r/ 70.5%

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

      6. associate-*l/ 70.5%

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

      7. *-un-lft-identity 70.5%

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

      8. associate-*l/ 69.9%

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

      9. *-commutative 69.9%

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

      10. associate-*l/ 70.5%

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

      11. *-un-lft-identity 70.5%

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

      12. associate-*r/ 70.5%

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

      13. associate-/r/ 70.5%

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

      14. /-rgt-identity 70.5%

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

   9. Applied egg-rr 70.5%

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

   10. Taylor expanded in m around 0 70.5%

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

   if 3.1000000000000001e-103 < 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. Add Preprocessing

   5. Taylor expanded in m around 0 66.6%

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

      1. +-commutative 66.6%

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

      2. distribute-lft-in 66.6%

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

      3. div-inv 66.7%

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

      4. *-rgt-identity 66.7%

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

   7. Applied egg-rr 66.7%

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

   8. Taylor expanded in m around inf 63.4%

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

      1. distribute-rgt-in 63.4%

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

      2. *-lft-identity 63.4%

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

      3. associate-*l/ 63.5%

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

      4. *-lft-identity 63.5%

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

   10. Simplified 63.5%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-164}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 4.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{-103}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} 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 <= 1.6d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (m / v))
    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 <= 1.6) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = (m / v) * (m * (m + -2.0));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   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* 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. Add Preprocessing

   5. Taylor expanded in m around 0 98.7%

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

   if 1.6000000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-*l/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf 98.4%

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

      1. +-commutative 98.4%

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

      2. unpow2 98.4%

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

      3. distribute-rgt-out 98.4%

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

   10. Simplified 98.4%

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

4. Final simplification 98.5%

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

Alternative 7: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.8 \cdot 10^{-166}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 9.8 \cdot 10^{-129} \lor \neg \left(m \leq 3.1 \cdot 10^{-103}\right):\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 8.8e-166)
   -1.0
   (if (or (<= m 9.8e-129) (not (<= m 3.1e-103))) (/ m v) -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.8e-166) {
		tmp = -1.0;
	} else if ((m <= 9.8e-129) || !(m <= 3.1e-103)) {
		tmp = m / v;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 8.8d-166) then
        tmp = -1.0d0
    else if ((m <= 9.8d-129) .or. (.not. (m <= 3.1d-103))) then
        tmp = m / v
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 8.8e-166) {
		tmp = -1.0;
	} else if ((m <= 9.8e-129) || !(m <= 3.1e-103)) {
		tmp = m / v;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8.8e-166:
		tmp = -1.0
	elif (m <= 9.8e-129) or not (m <= 3.1e-103):
		tmp = m / v
	else:
		tmp = -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8.8e-166)
		tmp = -1.0;
	elseif ((m <= 9.8e-129) || !(m <= 3.1e-103))
		tmp = Float64(m / v);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 8.8e-166)
		tmp = -1.0;
	elseif ((m <= 9.8e-129) || ~((m <= 3.1e-103)))
		tmp = m / v;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 8.8e-166], -1.0, If[Or[LessEqual[m, 9.8e-129], N[Not[LessEqual[m, 3.1e-103]], $MachinePrecision]], N[(m / v), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if m < 8.8000000000000005e-166 or 9.80000000000000004e-129 < m < 3.1000000000000001e-103

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 86.8%

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

   if 8.8000000000000005e-166 < m < 9.80000000000000004e-129 or 3.1000000000000001e-103 < 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. Add Preprocessing

   5. Taylor expanded in v around 0 95.3%

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

      1. associate-*l/ 95.3%

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

   7. Simplified 95.3%

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

      1. associate-*l/ 95.3%

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

      2. unpow2 95.3%

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

      3. associate-*r* 95.3%

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

      4. /-rgt-identity 95.3%

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

      5. associate-/r/ 95.3%

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

      6. associate-*l/ 95.3%

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

      7. *-un-lft-identity 95.3%

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

      8. associate-*l/ 95.2%

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

      9. *-commutative 95.2%

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

      10. associate-*l/ 95.3%

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

      11. *-un-lft-identity 95.3%

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

      12. associate-*r/ 95.3%

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

      13. associate-/r/ 95.3%

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

      14. /-rgt-identity 95.3%

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

   9. Applied egg-rr 95.3%

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

   10. Taylor expanded in m around 0 63.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.8 \cdot 10^{-166}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 9.8 \cdot 10^{-129} \lor \neg \left(m \leq 3.1 \cdot 10^{-103}\right):\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in m around 0 76.2%

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

   1. +-commutative 76.2%

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

   2. distribute-lft-in 76.2%

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

   3. div-inv 76.3%

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

   4. *-rgt-identity 76.3%

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

7. Applied egg-rr 76.3%

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

8. Final simplification 76.3%

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

Alternative 9: 27.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9 \cdot 10^{-45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 9e-45) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 9e-45) {
		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 <= 9d-45) 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 <= 9e-45) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 9e-45:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 9e-45)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 9e-45)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 9e-45], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 8.9999999999999997e-45

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 63.8%

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

   if 8.9999999999999997e-45 < 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. Add Preprocessing

   5. Taylor expanded in m around 0 61.1%

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

   6. Taylor expanded in m around inf 61.1%

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

   7. Taylor expanded in v around inf 5.7%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9 \cdot 10^{-45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in v around inf 28.2%

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

   1. neg-mul-1 28.2%

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

   2. neg-sub0 28.2%

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

   3. associate--r- 28.2%

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

   4. metadata-eval 28.2%

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

7. Simplified 28.2%

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

8. Final simplification 28.2%

   \[\leadsto m + -1 \]
9. Add Preprocessing

Alternative 11: 25.0% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in m around 0 25.4%

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

6. Final simplification 25.4%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))