b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.1s

Alternatives: 15

Speedup: 1.0×

• 41.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

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 2: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 96.5%

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

      1. sub-neg 96.5%

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

      2. distribute-rgt-in 96.5%

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

      3. *-un-lft-identity 96.5%

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

   5. Applied egg-rr 96.5%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/r/ 99.8%

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

      5. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around inf 99.3%

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

      1. neg-mul-1 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 98.0%

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

Alternative 3: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 96.5%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-/r/ 99.8%

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

      5. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around inf 99.3%

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

      1. neg-mul-1 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 98.0%

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

Alternative 4: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.43) (+ (/ (* m (- 1.0 m)) v) -1.0) (* m (+ 1.0 (/ (* m m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = ((m * (1.0 - m)) / v) + -1.0;
	} else {
		tmp = m * (1.0 + ((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 <= 0.43d0) then
        tmp = ((m * (1.0d0 - m)) / v) + (-1.0d0)
    else
        tmp = m * (1.0d0 + ((m * m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = ((m * (1.0 - m)) / v) + -1.0;
	} else {
		tmp = m * (1.0 + ((m * m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.43:
		tmp = ((m * (1.0 - m)) / v) + -1.0
	else:
		tmp = m * (1.0 + ((m * m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.43)
		tmp = Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0);
	else
		tmp = Float64(m * Float64(1.0 + Float64(Float64(m * m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.43)
		tmp = ((m * (1.0 - m)) / v) + -1.0;
	else
		tmp = m * (1.0 + ((m * m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.43], N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(m * N[(1.0 + N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.429999999999999993

   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. Taylor expanded in m around 0 96.5%

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

   if 0.429999999999999993 < m

   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. Taylor expanded in m around inf 99.3%

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

      1. neg-mul-1 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in m around inf 99.2%

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

      1. neg-mul-1 99.3%

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

   8. Simplified 99.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.43) (* (- 1.0 m) (+ -1.0 (/ m v))) (* m (+ 1.0 (/ (* m m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * (1.0 + ((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 <= 0.43d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (m / v))
    else
        tmp = m * (1.0d0 + ((m * m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * (1.0 + ((m * m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.43:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = m * (1.0 + ((m * m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.43)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(1.0 + Float64(Float64(m * m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.43)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	else
		tmp = m * (1.0 + ((m * m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.43], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(1.0 + N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.429999999999999993

   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. Taylor expanded in m around 0 96.5%

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

   if 0.429999999999999993 < m

   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. Taylor expanded in m around inf 99.3%

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

      1. neg-mul-1 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in m around inf 99.2%

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

      1. neg-mul-1 99.3%

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

   8. Simplified 99.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(N[(1.0 - m), $MachinePrecision] / N[(v / m), $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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. div-inv 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-/r/ 99.8%

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

   5. un-div-inv 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 7: 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{-180}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3.5e-180) -1.0 (+ m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.5e-180) {
		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 <= 3.5d-180) 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 <= 3.5e-180) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.5e-180:
		tmp = -1.0
	else:
		tmp = m + (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.5e-180)
		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 <= 3.5e-180)
		tmp = -1.0;
	else
		tmp = m + (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.5e-180], -1.0, N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.5000000000000001e-180

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 85.2%

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

   if 3.5000000000000001e-180 < 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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-*r/ 99.9%

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

      5. div-inv 99.8%

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

      6. associate-*r* 99.8%

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

      7. fma-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. fmm-undef 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in m around 0 69.9%

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

   10. Taylor expanded in m around inf 58.7%

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

       1. distribute-rgt-in 58.7%

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

       2. *-lft-identity 58.7%

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

       3. associate-*l/ 58.8%

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

       4. *-lft-identity 58.8%

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

   12. Simplified 58.8%

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

Alternative 9: 87.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(m + 1.0), $MachinePrecision] * 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. Add Preprocessing

3. Taylor expanded in m around 0 43.8%

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

   1. sub-neg 43.8%

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

   2. distribute-lft-in 43.8%

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

   3. *-commutative 43.8%

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

   4. *-un-lft-identity 43.8%

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

   5. sub-neg 43.8%

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

   6. metadata-eval 43.8%

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

   7. sub-neg 43.8%

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

   8. metadata-eval 43.8%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 85.0%

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

   11. sqr-neg 85.0%

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

   12. sqrt-unprod 85.0%

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

   13. add-sqr-sqrt 85.0%

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

5. Applied egg-rr 85.0%

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

   1. *-commutative 85.0%

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

   2. distribute-rgt1-in 85.0%

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

7. Simplified 85.0%

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

8. Final simplification 85.0%

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

Alternative 10: 62.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{-180}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3.5e-180) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.5e-180) {
		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 <= 3.5d-180) 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 <= 3.5e-180) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.5e-180:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.5e-180)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.5e-180)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.5e-180], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.5000000000000001e-180

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 85.2%

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

   if 3.5000000000000001e-180 < 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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-*r/ 99.9%

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

      5. div-inv 99.8%

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

      6. associate-*r* 99.8%

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

      7. fma-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. fmm-undef 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in m around 0 69.9%

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

   10. Taylor expanded in v around 0 58.8%

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

Alternative 11: 76.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. distribute-lft-in 99.8%

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

   2. *-commutative 99.8%

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

   3. neg-mul-1 99.8%

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

   4. associate-*r/ 99.9%

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

   5. div-inv 99.8%

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

   6. associate-*r* 99.8%

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

   7. fma-define 99.8%

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

6. Applied egg-rr 99.8%

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

   1. fmm-undef 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in m around 0 74.6%

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

10. Final simplification 74.6%

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

Alternative 12: 27.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.9e-19) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-19) {
		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 <= 2.9d-19) 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 <= 2.9e-19) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.9e-19:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.9e-19)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.9e-19)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.9e-19], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 2.9e-19

   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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 52.3%

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

   if 2.9e-19 < m

   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. Taylor expanded in m around inf 92.7%

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

      1. neg-mul-1 92.7%

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

   5. Simplified 92.7%

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

   6. Taylor expanded in m around 0 5.1%

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

   7. Taylor expanded in m around inf 5.4%

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

Alternative 13: 76.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return -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 / v)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(-1.0 + N[(m / v), $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. Taylor expanded in m around 0 43.8%

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

4. Taylor expanded in m around 0 74.6%

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

5. Final simplification 74.6%

   \[\leadsto -1 + \frac{m}{v} \]
6. Add Preprocessing

Alternative 14: 27.7% 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 24.7%

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

   1. neg-mul-1 24.7%

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

   2. neg-sub0 24.7%

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

   3. associate--r- 24.7%

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

   4. metadata-eval 24.7%

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

7. Simplified 24.7%

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

8. Final simplification 24.7%

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

Alternative 15: 25.3% 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 22.1%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(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)))