b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 10.2s

Alternatives: 13

Speedup: 1.0×

• 41.1% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(N[(N[(N[(m * N[(1.0 - N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 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. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. flip-- 100.0%

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

   3. associate-*l/ 100.0%

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

   4. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{v}{m - m \cdot m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 8.5e-17) (+ m (+ -1.0 (/ m v))) (/ (- 1.0 m) (/ v (- m (* m m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.5e-17) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) / (v / (m - (m * m)));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 8.5d-17) then
        tmp = m + ((-1.0d0) + (m / v))
    else
        tmp = (1.0d0 - m) / (v / (m - (m * m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 8.5e-17) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) / (v / (m - (m * m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8.5e-17:
		tmp = m + (-1.0 + (m / v))
	else:
		tmp = (1.0 - m) / (v / (m - (m * m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8.5e-17)
		tmp = Float64(m + Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) / Float64(v / Float64(m - Float64(m * m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 8.5e-17)
		tmp = m + (-1.0 + (m / v));
	else
		tmp = (1.0 - m) / (v / (m - (m * m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 8.5e-17], N[(m + N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] / N[(v / N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.5e-17

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. *-lft-identity 99.8%

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

      3. associate--l+ 99.8%

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

      4. associate-*l/ 100.0%

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

      5. *-lft-identity 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if 8.5e-17 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. unpow2 99.9%

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

   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%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{v}{m - m \cdot m}}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(N[(m - N[(m * 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. Taylor expanded in m around 0 100.0%

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

   1. mul-1-neg 100.0%

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

   2. unsub-neg 100.0%

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

   3. unpow2 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 6: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 97.6%

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

   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 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. associate-/l* 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in v around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. associate-*l/ 99.1%

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

      4. associate-*r/ 99.0%

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

      5. *-commutative 99.0%

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

   9. Simplified 99.0%

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

4. Final simplification 98.3%

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

Alternative 7: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 97.6%

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

   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 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. associate-/l* 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in v around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. *-commutative 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 98.3%

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

Alternative 8: 62.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.2 \cdot 10^{-159}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 8.2e-159) -1.0 (+ m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.2e-159) {
		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 <= 8.2d-159) 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 <= 8.2e-159) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8.2e-159:
		tmp = -1.0
	else:
		tmp = m + (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8.2e-159)
		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 <= 8.2e-159)
		tmp = -1.0;
	else
		tmp = m + (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 8.2e-159], -1.0, N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.20000000000000029e-159

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 79.6%

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

   if 8.20000000000000029e-159 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 65.1%

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

      1. distribute-rgt-in 65.1%

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

      2. *-lft-identity 65.1%

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

      3. associate--l+ 65.1%

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

      4. associate-*l/ 65.2%

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

      5. *-lft-identity 65.2%

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

      6. sub-neg 65.2%

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

      7. metadata-eval 65.2%

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

   6. Simplified 65.2%

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

   7. Taylor expanded in m around inf 56.1%

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

      1. distribute-lft-in 56.1%

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

      2. /-rgt-identity 56.1%

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

      3. times-frac 56.3%

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

      4. *-rgt-identity 56.3%

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

      5. *-lft-identity 56.3%

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

      6. *-rgt-identity 56.3%

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

   9. Simplified 56.3%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.2 \cdot 10^{-159}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \]

Alternative 9: 75.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(m + N[(-1.0 + 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. Taylor expanded in m around 0 73.6%

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

   1. distribute-rgt-in 73.6%

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

   2. *-lft-identity 73.6%

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

   3. associate--l+ 73.6%

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

   4. associate-*l/ 73.7%

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

   5. *-lft-identity 73.7%

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

   6. sub-neg 73.7%

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

   7. metadata-eval 73.7%

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

6. Simplified 73.7%

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

7. Final simplification 73.7%

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

Alternative 10: 62.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.6 \cdot 10^{-160}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 6.6e-160) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.6e-160) {
		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 <= 6.6d-160) 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 <= 6.6e-160) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.6e-160:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.6e-160)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.6e-160)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.6e-160], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.6e-160

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 79.6%

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

   if 6.6e-160 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. unpow2 99.9%

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

   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 91.0%

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

      1. associate-/l* 91.0%

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

      2. unpow2 91.0%

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

   7. Simplified 91.0%

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

      1. div-inv 90.9%

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

      2. *-un-lft-identity 90.9%

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

      3. distribute-rgt-out-- 90.9%

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

   9. Applied egg-rr 90.9%

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

   10. Taylor expanded in m around 0 56.3%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.6 \cdot 10^{-160}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 11: 26.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8 \cdot 10^{-55}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 8e-55) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8e-55) {
		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 <= 8d-55) 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 <= 8e-55) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8e-55:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8e-55)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 8e-55)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 8e-55], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 7.99999999999999996e-55

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 60.4%

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

   if 7.99999999999999996e-55 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 11.5%

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

   3. Taylor expanded in m around inf 10.8%

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

      1. +-commutative 10.8%

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

      2. distribute-rgt-in 10.8%

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

      3. *-lft-identity 10.8%

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

      4. mul-1-neg 10.8%

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

      5. unpow2 10.8%

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

      6. associate-+l+ 10.8%

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

      7. associate-*l/ 10.8%

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

      8. *-lft-identity 10.8%

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

      9. sub-neg 10.8%

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

      10. *-rgt-identity 10.8%

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

      11. associate-*l/ 10.8%

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

      12. distribute-lft-out-- 10.9%

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

      13. associate-*l/ 10.8%

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

      14. /-rgt-identity 10.8%

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

      15. *-lft-identity 10.8%

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

      16. *-commutative 10.8%

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

      17. associate-*r/ 10.8%

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

      18. /-rgt-identity 10.8%

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

      19. associate-*l* 10.8%

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

      20. associate-*r/ 10.8%

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

      21. *-rgt-identity 10.8%

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

   5. Simplified 10.8%

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

   6. Taylor expanded in v around inf 5.7%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8 \cdot 10^{-55}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]

Alternative 12: 26.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around inf 28.5%

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

   1. neg-mul-1 28.5%

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

   2. neg-sub0 28.5%

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

   3. associate--r- 28.5%

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

   4. metadata-eval 28.5%

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

6. Simplified 28.5%

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

7. Final simplification 28.5%

   \[\leadsto m + -1 \]

Alternative 13: 24.3% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 25.9%

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

5. Final simplification 25.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023283 
(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)))