b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 13

Speedup: 1.0×

• 40.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. distribute-rgt-in 99.9%

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

   3. *-un-lft-identity 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;-1 + m \cdot \frac{1 + m \cdot -2}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 - m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.62)
   (+ -1.0 (* m (/ (+ 1.0 (* m -2.0)) v)))
   (* (- 1.0 m) (- -1.0 (* m (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.62) {
		tmp = -1.0 + (m * ((1.0 + (m * -2.0)) / v));
	} 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 <= 0.62d0) then
        tmp = (-1.0d0) + (m * ((1.0d0 + (m * (-2.0d0))) / v))
    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 <= 0.62) {
		tmp = -1.0 + (m * ((1.0 + (m * -2.0)) / v));
	} else {
		tmp = (1.0 - m) * (-1.0 - (m * (m / v)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.62:
		tmp = -1.0 + (m * ((1.0 + (m * -2.0)) / v))
	else:
		tmp = (1.0 - m) * (-1.0 - (m * (m / v)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.62)
		tmp = Float64(-1.0 + Float64(m * Float64(Float64(1.0 + Float64(m * -2.0)) / v)));
	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 <= 0.62)
		tmp = -1.0 + (m * ((1.0 + (m * -2.0)) / v));
	else
		tmp = (1.0 - m) * (-1.0 - (m * (m / v)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.62], N[(-1.0 + N[(m * N[(N[(1.0 + N[(m * -2.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $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 < 0.619999999999999996

   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 99.4%

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

   6. Taylor expanded in v around 0 99.6%

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

      1. associate-/l* 99.5%

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

      2. *-commutative 99.5%

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

   8. Simplified 99.5%

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

   if 0.619999999999999996 < 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 97.5%

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

      1. neg-mul-1 97.5%

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

      2. distribute-neg-frac2 97.5%

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

   7. Simplified 97.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;-1 + m \cdot \frac{1 + m \cdot -2}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 - m \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   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 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   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. clear-num 99.8%

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

   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 5: 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 6: 88.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in m around 0 49.1%

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

   1. *-commutative 49.1%

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

   2. sub-neg 49.1%

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

   3. metadata-eval 49.1%

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

   4. distribute-rgt-in 49.1%

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

   5. sub-neg 49.1%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 86.5%

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

   8. sqr-neg 86.5%

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

   9. sqrt-unprod 86.5%

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

   10. add-sqr-sqrt 86.5%

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

   11. sub-neg 86.5%

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

   12. distribute-lft-in 86.5%

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

   13. metadata-eval 86.5%

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

   14. add-sqr-sqrt 0.0%

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

   15. sqrt-unprod 61.1%

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

   16. sqr-neg 61.1%

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

   17. sqrt-unprod 86.5%

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

   18. add-sqr-sqrt 86.5%

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

   19. neg-mul-1 86.5%

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

   20. add-sqr-sqrt 0.0%

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

   21. sqrt-unprod 86.5%

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

   22. sqr-neg 86.5%

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

   23. sqrt-unprod 86.5%

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

   24. add-sqr-sqrt 86.5%

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

5. Applied egg-rr 86.5%

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

6. Final simplification 86.5%

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

Alternative 7: 88.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(-1.0 + N[(m + N[(N[(m * N[(m + 1.0), $MachinePrecision]), $MachinePrecision] / 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 49.1%

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

   1. *-commutative 49.1%

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

   2. sub-neg 49.1%

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

   3. metadata-eval 49.1%

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

   4. distribute-rgt-in 49.1%

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

   5. sub-neg 49.1%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 86.5%

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

   8. sqr-neg 86.5%

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

   9. sqrt-unprod 86.5%

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

   10. add-sqr-sqrt 86.5%

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

   11. sub-neg 86.5%

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

   12. distribute-lft-in 86.5%

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

   13. metadata-eval 86.5%

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

   14. add-sqr-sqrt 0.0%

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

   15. sqrt-unprod 61.1%

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

   16. sqr-neg 61.1%

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

   17. sqrt-unprod 86.5%

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

   18. add-sqr-sqrt 86.5%

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

   19. neg-mul-1 86.5%

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

   20. add-sqr-sqrt 0.0%

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

   21. sqrt-unprod 86.5%

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

   22. sqr-neg 86.5%

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

   23. sqrt-unprod 86.5%

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

   24. add-sqr-sqrt 86.5%

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

5. Applied egg-rr 86.5%

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

6. Taylor expanded in v around inf 86.5%

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

7. Final simplification 86.5%

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

Alternative 8: 88.1% 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 49.1%

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

   1. sub-neg 49.1%

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

   2. distribute-rgt-in 49.1%

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

   3. *-un-lft-identity 49.1%

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

   4. sub-neg 49.1%

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

   5. metadata-eval 49.1%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 86.5%

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

   8. sqr-neg 86.5%

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

   9. sqrt-unprod 86.5%

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

   10. add-sqr-sqrt 86.5%

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

   11. sub-neg 86.5%

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

   12. metadata-eval 86.5%

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

5. Applied egg-rr 86.5%

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

   1. distribute-rgt1-in 86.5%

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

7. Simplified 86.5%

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

8. Final simplification 86.5%

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

Alternative 9: 88.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(-1.0 + N[(N[(m * N[(m + 1.0), $MachinePrecision]), $MachinePrecision] / 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 49.1%

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

   1. *-commutative 49.1%

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

   2. sub-neg 49.1%

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

   3. metadata-eval 49.1%

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

   4. distribute-rgt-in 49.1%

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

   5. sub-neg 49.1%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 86.5%

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

   8. sqr-neg 86.5%

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

   9. sqrt-unprod 86.5%

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

   10. add-sqr-sqrt 86.5%

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

   11. sub-neg 86.5%

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

   12. distribute-lft-in 86.5%

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

   13. metadata-eval 86.5%

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

   14. add-sqr-sqrt 0.0%

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

   15. sqrt-unprod 61.1%

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

   16. sqr-neg 61.1%

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

   17. sqrt-unprod 86.5%

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

   18. add-sqr-sqrt 86.5%

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

   19. neg-mul-1 86.5%

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

   20. add-sqr-sqrt 0.0%

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

   21. sqrt-unprod 86.5%

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

   22. sqr-neg 86.5%

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

   23. sqrt-unprod 86.5%

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

   24. add-sqr-sqrt 86.5%

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

5. Applied egg-rr 86.5%

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

6. Taylor expanded in v around inf 86.5%

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

7. Taylor expanded in v around 0 86.5%

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

8. Final simplification 86.5%

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

Alternative 10: 62.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 6e-184) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6e-184) {
		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 <= 6d-184) 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 <= 6e-184) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6e-184:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6e-184)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6e-184)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6e-184], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.99999999999999982e-184

   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 79.6%

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

   if 5.99999999999999982e-184 < 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 36.1%

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

   4. Taylor expanded in v around 0 26.7%

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

   5. Taylor expanded in m around 0 59.6%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   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 49.4%

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

6. Taylor expanded in m around 0 75.3%

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

   1. distribute-rgt-in 75.3%

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

   2. *-lft-identity 75.3%

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

   3. associate-*l/ 75.3%

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

   4. *-lft-identity 75.3%

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

8. Simplified 75.3%

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

9. Final simplification 75.3%

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

Alternative 12: 27.8% 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 26.7%

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

   1. neg-mul-1 26.7%

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

   2. sub-neg 26.7%

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

   3. +-commutative 26.7%

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

   4. distribute-neg-in 26.7%

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

   5. remove-double-neg 26.7%

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

   6. metadata-eval 26.7%

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

7. Simplified 26.7%

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

8. Final simplification 26.7%

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

Alternative 13: 25.4% 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 24.2%

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

6. Final simplification 24.2%

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

Reproduce

herbie shell --seed 2024075 
(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)))