b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.7s

Alternatives: 11

Speedup: N/A×

• 41.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(m + \frac{m \cdot \left(1 + m \cdot \left(m + -2\right)\right)}{v}\right) + -1 \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (+ (+ m (/ (* m (+ 1.0 (* m (+ m -2.0)))) v)) -1.0))

C

double code(double m, double v) {
	return (m + ((m * (1.0 + (m * (m + -2.0)))) / v)) + -1.0;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (m + ((m * (1.0d0 + (m * (m + (-2.0d0))))) / v)) + (-1.0d0)
end function

Java

public static double code(double m, double v) {
	return (m + ((m * (1.0 + (m * (m + -2.0)))) / v)) + -1.0;
}

Python

def code(m, v):
	return (m + ((m * (1.0 + (m * (m + -2.0)))) / v)) + -1.0

Julia

function code(m, v)
	return Float64(Float64(m + Float64(Float64(m * Float64(1.0 + Float64(m * Float64(m + -2.0)))) / v)) + -1.0)
end

MATLAB

function tmp = code(m, v)
	tmp = (m + ((m * (1.0 + (m * (m + -2.0)))) / v)) + -1.0;
end

Wolfram

code[m_, v_] := N[(N[(m + N[(N[(m * N[(1.0 + N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $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. 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* 99.5%

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

   4. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. distribute-rgt-in 99.5%

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

   2. associate-*r/ 99.9%

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

   3. clear-num 99.8%

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

   4. associate-*l/ 99.8%

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

   5. *-un-lft-identity 99.8%

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

   6. associate-/r* 99.8%

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

   7. neg-mul-1 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in v around 0 99.9%

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

8. Taylor expanded in m around 0 99.9%

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

   1. unpow2 99.9%

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

   2. distribute-rgt-out 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (1.0 - m) * ((m / v) + -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[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 - N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $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 98.1%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      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 inf 96.4%

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

      1. neg-mul-1 96.4%

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

      2. distribute-neg-frac2 96.4%

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

   7. Simplified 96.4%

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

4. Final simplification 97.3%

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

Alternative 3: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * ((m / v) + -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 = (1.0d0 - m) * ((m / v) + (-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 = (1.0 - m) * ((m / v) + -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 = (1.0 - m) * ((m / v) + -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(1.0 - m) * Float64(Float64(m / v) + -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 = (1.0 - m) * ((m / v) + -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[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $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 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 98.1%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      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 inf 96.4%

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

      1. neg-mul-1 96.4%

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

      2. distribute-neg-frac2 96.4%

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

   7. Simplified 96.4%

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

      1. *-commutative 96.4%

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

      2. distribute-frac-neg2 96.4%

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

      3. distribute-frac-neg 96.4%

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

      4. associate-*l/ 96.5%

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

   9. Applied egg-rr 96.5%

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

4. Final simplification 97.3%

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

Alternative 4: 87.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m + -1.0), $MachinePrecision] * N[(N[(m / v), $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. 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* 99.0%

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

      4. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in m around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. distribute-lft-in 97.1%

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

      3. div-inv 98.0%

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

      4. *-rgt-identity 98.0%

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

   7. Applied egg-rr 98.0%

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

   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 0 0.1%

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

      1. pow1 0.1%

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

      2. *-commutative 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 68.6%

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

      5. sqr-neg 68.6%

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

      6. sqrt-unprod 68.6%

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

      7. add-sqr-sqrt 68.6%

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

      8. neg-sub0 68.6%

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

      9. associate--r- 68.6%

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

      10. metadata-eval 68.6%

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

      11. sub-neg 68.6%

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

      12. metadata-eval 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. unpow1 68.6%

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

      2. +-commutative 68.6%

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

   7. Simplified 68.6%

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

4. Final simplification 83.4%

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

Alternative 5: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{m}{v} + -1\\ \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(m + -1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (+ (/ m v) -1.0)))
   (if (<= m 1.0) (* (- 1.0 m) t_0) (* (+ m -1.0) t_0))))

C

double code(double m, double v) {
	double t_0 = (m / v) + -1.0;
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * t_0;
	} else {
		tmp = (m + -1.0) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m / v) + (-1.0d0)
    if (m <= 1.0d0) then
        tmp = (1.0d0 - m) * t_0
    else
        tmp = (m + (-1.0d0)) * t_0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = (m / v) + -1.0;
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * t_0;
	} else {
		tmp = (m + -1.0) * t_0;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = (m / v) + -1.0
	tmp = 0
	if m <= 1.0:
		tmp = (1.0 - m) * t_0
	else:
		tmp = (m + -1.0) * t_0
	return tmp

Julia

function code(m, v)
	t_0 = Float64(Float64(m / v) + -1.0)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(1.0 - m) * t_0);
	else
		tmp = Float64(Float64(m + -1.0) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = (m / v) + -1.0;
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (1.0 - m) * t_0;
	else
		tmp = (m + -1.0) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(m + -1.0), $MachinePrecision] * t$95$0), $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 98.1%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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. pow1 0.1%

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

      2. *-commutative 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 68.6%

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

      5. sqr-neg 68.6%

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

      6. sqrt-unprod 68.6%

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

      7. add-sqr-sqrt 68.6%

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

      8. neg-sub0 68.6%

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

      9. associate--r- 68.6%

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

      10. metadata-eval 68.6%

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

      11. sub-neg 68.6%

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

      12. metadata-eval 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. unpow1 68.6%

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

      2. +-commutative 68.6%

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

   7. Simplified 68.6%

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

4. Final simplification 83.5%

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

Alternative 6: 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* 99.5%

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

   4. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.5%

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

   4. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.4%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 8: 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/ 99.9%

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

   3. *-commutative 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 9: 76.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.5%

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

   4. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in m around 0 70.6%

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

   1. +-commutative 70.6%

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

   2. distribute-lft-in 70.6%

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

   3. div-inv 71.1%

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

   4. *-rgt-identity 71.1%

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

7. Applied egg-rr 71.1%

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

8. Final simplification 71.1%

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

Alternative 10: 27.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.5%

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

   4. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in v around inf 23.7%

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

   1. neg-mul-1 23.7%

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

   2. neg-sub0 23.7%

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

   3. associate--r- 23.7%

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

   4. metadata-eval 23.7%

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

7. Simplified 23.7%

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

8. Final simplification 23.7%

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

Alternative 11: 24.9% 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* 99.5%

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

   4. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in m around 0 21.6%

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

6. Final simplification 21.6%

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

Reproduce

herbie shell --seed 2024040 
(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)))