b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.2s

Alternatives: 12

Speedup: N/A×

• 42.2% 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

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

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(Float64(m / 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[(N[(m / v), $MachinePrecision] * N[(1.0 - 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. Add Preprocessing
3. 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. associate-*r/ 100.0%

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

   3. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in m around 0 97.3%

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

   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. Add Preprocessing

   3. Taylor expanded in m around inf 97.8%

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

      1. neg-mul-1 97.8%

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

   5. Simplified 97.8%

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

4. Final simplification 97.6%

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

Alternative 3: 97.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0)
   (+ (* (/ m v) (- 1.0 m)) -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 / v) * (1.0 - m)) + -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 / v) * (1.0d0 - m)) + (-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 / v) * (1.0 - m)) + -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 / v) * (1.0 - m)) + -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 / v) * Float64(1.0 - m)) + -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 / v) * (1.0 - m)) + -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 / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $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 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in m around 0 97.3%

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

   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* 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 99.9%

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

      1. +-commutative 99.9%

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

      2. distribute-rgt-in 50.7%

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

      3. associate-/r/ 50.7%

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

      4. unpow-1 50.7%

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

      5. neg-mul-1 50.7%

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

      6. distribute-lft-neg-out 50.7%

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

      7. associate-/r/ 50.7%

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

      8. sub-neg 50.7%

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

      9. unpow-1 50.7%

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

      10. div-sub 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 97.8%

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

      1. neg-mul-1 97.8%

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

   10. Simplified 97.8%

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

4. Final simplification 97.6%

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

Alternative 4: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $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 < 1

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in m around 0 97.3%

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

   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. Add Preprocessing

   3. Taylor expanded in m around inf 97.8%

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

      1. neg-mul-1 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in m around inf 97.7%

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

      1. neg-mul-1 97.8%

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

   8. Simplified 97.7%

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

4. Final simplification 97.5%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 6: 88.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (+ (/ m v) -1.0) (* m (/ (+ m 1.0) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = 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 <= 2.4d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = m * ((m + 1.0d0) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = (m / v) + -1.0
	else:
		tmp = m * ((m + 1.0) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(m * Float64(Float64(m + 1.0) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = (m / v) + -1.0;
	else
		tmp = m * ((m + 1.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(m * N[(N[(m + 1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   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%

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

   4. Taylor expanded in m around 0 96.6%

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

   5. Taylor expanded in m around 0 96.6%

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

   if 2.39999999999999991 < 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 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-lft-in 0.1%

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

      5. associate-*r/ 0.1%

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

      6. sub-neg 0.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 79.1%

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

      9. sqr-neg 79.1%

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

      10. sqrt-unprod 79.1%

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

      11. add-sqr-sqrt 79.1%

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

      12. +-commutative 79.1%

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

      13. distribute-lft1-in 79.1%

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

      14. fma-define 79.1%

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

      15. sub-neg 79.1%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 33.3%

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

      18. sqr-neg 33.3%

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

      19. sqrt-unprod 79.1%

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

      20. add-sqr-sqrt 79.1%

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

      21. +-commutative 79.1%

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

   5. Applied egg-rr 79.1%

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

      1. fma-undefine 79.1%

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

      2. distribute-lft1-in 79.1%

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

      3. associate-*r/ 79.1%

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

      4. distribute-lft-in 79.1%

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

   7. Simplified 79.1%

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

   8. Taylor expanded in v around 0 79.1%

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

      1. associate-*r/ 79.1%

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

      2. +-commutative 79.1%

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

   10. Simplified 79.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(m + 1.0), $MachinePrecision] * N[(N[(m / 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. Taylor expanded in m around 0 47.2%

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

   1. *-commutative 47.2%

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

   2. sub-neg 47.2%

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

   3. metadata-eval 47.2%

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

   4. distribute-lft-in 47.2%

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

   5. associate-*r/ 47.2%

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

   6. sub-neg 47.2%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 87.6%

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

   9. sqr-neg 87.6%

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

   10. sqrt-unprod 87.6%

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

   11. add-sqr-sqrt 87.6%

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

   12. +-commutative 87.6%

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

   13. distribute-lft1-in 87.6%

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

   14. fma-define 87.6%

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

   15. sub-neg 87.6%

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

   16. add-sqr-sqrt 0.0%

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

   17. sqrt-unprod 64.1%

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

   18. sqr-neg 64.1%

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

   19. sqrt-unprod 87.6%

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

   20. add-sqr-sqrt 87.6%

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

   21. +-commutative 87.6%

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

5. Applied egg-rr 87.6%

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

   1. fma-undefine 87.6%

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

   2. distribute-lft1-in 87.6%

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

   3. associate-*r/ 87.6%

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

   4. distribute-lft-in 87.6%

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

7. Simplified 87.6%

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

Alternative 8: 58.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 7 \cdot 10^{-106}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 7e-106) (/ m v) (+ m -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 7e-106) {
		tmp = m / v;
	} else {
		tmp = 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 (v <= 7d-106) then
        tmp = m / v
    else
        tmp = m + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 7e-106) {
		tmp = m / v;
	} else {
		tmp = m + -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 7e-106:
		tmp = m / v
	else:
		tmp = m + -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 7e-106)
		tmp = Float64(m / v);
	else
		tmp = Float64(m + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 7e-106)
		tmp = m / v;
	else
		tmp = m + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 7e-106], N[(m / v), $MachinePrecision], N[(m + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 7e-106

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 47.3%

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

   4. Taylor expanded in m around 0 84.3%

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

   5. Taylor expanded in m around inf 72.9%

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

   if 7e-106 < v

   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. Taylor expanded in v around inf 46.5%

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

      1. neg-mul-1 46.5%

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

      2. sub-neg 46.5%

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

      3. +-commutative 46.5%

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

      4. distribute-neg-in 46.5%

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

      5. remove-double-neg 46.5%

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

      6. metadata-eval 46.5%

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

   7. Simplified 46.5%

      \[\leadsto \color{blue}{m + -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 26.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 10^{-17}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1e-17) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1e-17) {
		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 <= 1d-17) 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 <= 1e-17) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1e-17:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1e-17)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1e-17)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1e-17], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 1.00000000000000007e-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* 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. Taylor expanded in m around 0 49.9%

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

   if 1.00000000000000007e-17 < 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 91.7%

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

      1. neg-mul-1 91.7%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in m around 0 5.6%

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

   7. Taylor expanded in m around inf 5.7%

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

Alternative 10: 75.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in m around 0 47.2%

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

4. Taylor expanded in m around 0 74.4%

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

5. Taylor expanded in m around 0 74.4%

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

6. Final simplification 74.4%

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

Alternative 11: 26.6% 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%

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

   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. Taylor expanded in v around inf 25.5%

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

   1. neg-mul-1 25.5%

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

   2. sub-neg 25.5%

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

   3. +-commutative 25.5%

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

   4. distribute-neg-in 25.5%

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

   5. remove-double-neg 25.5%

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

   6. metadata-eval 25.5%

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

7. Simplified 25.5%

   \[\leadsto \color{blue}{m + -1} \]
8. Add Preprocessing

Alternative 12: 24.2% 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%

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

   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. Taylor expanded in m around 0 22.9%

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

Reproduce

herbie shell --seed 2024131 
(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)))