b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.7s

Alternatives: 11

Speedup: 1.0×

• 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, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return ((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 * (1.0d0 + (m * (m - 2.0d0)))) / v) + (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 99.8%

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

   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 99.8%

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

6. Taylor expanded in v around 0 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 87.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.65:\\ \;\;\;\;-1 + \frac{m \cdot \left(1 + m \cdot -2\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{v}{\left(m + -1\right) \cdot \left(m + v\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.65)
   (+ -1.0 (/ (* m (+ 1.0 (* m -2.0))) v))
   (/ 1.0 (/ v (* (+ m -1.0) (+ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.65) {
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	} else {
		tmp = 1.0 / (v / ((m + -1.0) * (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 <= 0.65d0) then
        tmp = (-1.0d0) + ((m * (1.0d0 + (m * (-2.0d0)))) / v)
    else
        tmp = 1.0d0 / (v / ((m + (-1.0d0)) * (m + v)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.65) {
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	} else {
		tmp = 1.0 / (v / ((m + -1.0) * (m + v)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.65:
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v)
	else:
		tmp = 1.0 / (v / ((m + -1.0) * (m + v)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.65)
		tmp = Float64(-1.0 + Float64(Float64(m * Float64(1.0 + Float64(m * -2.0))) / v));
	else
		tmp = Float64(1.0 / Float64(v / Float64(Float64(m + -1.0) * Float64(m + v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.65)
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	else
		tmp = 1.0 / (v / ((m + -1.0) * (m + v)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.65], N[(-1.0 + N[(N[(m * N[(1.0 + N[(m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(v / N[(N[(m + -1.0), $MachinePrecision] * N[(m + v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.650000000000000022

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 99.7%

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

   6. Taylor expanded in v around 0 100.0%

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

   7. Taylor expanded in m around 0 99.5%

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

      1. *-commutative 99.5%

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

   9. Simplified 99.5%

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

   if 0.650000000000000022 < 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. Taylor expanded in v around 0 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. neg-mul-1 0.1%

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

      4. distribute-rgt-out 0.1%

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

   6. Simplified 0.1%

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

      1. frac-2neg 0.1%

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

      2. div-inv 0.1%

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

      3. distribute-rgt-neg-in 0.1%

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

      4. unsub-neg 0.1%

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

      5. add-sqr-sqrt 0.1%

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

      6. sqrt-unprod 0.1%

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

      7. sqr-neg 0.1%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 0.1%

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

      10. neg-mul-1 0.1%

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

      11. cancel-sign-sub-inv 0.1%

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

      12. metadata-eval 0.1%

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

      13. *-un-lft-identity 0.1%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 83.6%

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

      16. sqr-neg 83.6%

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

      17. sqrt-unprod 81.5%

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

      18. add-sqr-sqrt 81.5%

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

   8. Applied egg-rr 81.5%

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

      1. associate-*l* 81.5%

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

      2. distribute-lft-neg-out 81.5%

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

      3. distribute-rgt-neg-in 81.5%

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

      4. distribute-lft-neg-in 81.5%

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

      5. associate-*r/ 81.5%

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

      6. *-rgt-identity 81.5%

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

      7. sub-neg 81.5%

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

      8. distribute-neg-in 81.5%

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

      9. metadata-eval 81.5%

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

      10. remove-double-neg 81.5%

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

   10. Simplified 81.5%

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

       1. associate-*r/ 81.5%

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

       2. clear-num 81.5%

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

   12. Applied egg-rr 81.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.65:\\ \;\;\;\;-1 + \frac{m \cdot \left(1 + m \cdot -2\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{v}{\left(m + -1\right) \cdot \left(m + v\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.8× speedup

Math

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

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(Float64(m * 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[(N[(m * N[(1.0 + N[(m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $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 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 99.7%

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

   6. Taylor expanded in v around 0 100.0%

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

   7. Taylor expanded in m around 0 99.5%

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

      1. *-commutative 99.5%

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

   9. Simplified 99.5%

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

   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* 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 inf 98.1%

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

      1. neg-mul-1 98.1%

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

      2. distribute-neg-frac2 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 98.9%

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

Alternative 4: 98.3% accurate, 0.8× speedup

Math

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

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 (/ v m))))))

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.619999999999999996

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 99.7%

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

   6. Taylor expanded in v around 0 100.0%

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

   7. Taylor expanded in m around 0 99.5%

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

      1. *-commutative 99.5%

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

   9. Simplified 99.5%

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

   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* 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. Step-by-step derivation

      1. clear-num 99.9%

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

      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. Taylor expanded in m around inf 98.1%

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

      1. neg-mul-1 98.1%

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

      2. distribute-neg-frac 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 98.9%

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

Alternative 5: 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 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* 99.8%

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

   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 6: 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 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* 99.8%

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

   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 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 7: 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 100.0%

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

3. Final simplification 100.0%

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

Alternative 8: 63.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.4 \cdot 10^{-150}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 7.4e-150) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.4e-150) {
		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 <= 7.4d-150) 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 <= 7.4e-150) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.4e-150:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.4e-150)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7.4e-150)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.4e-150], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.40000000000000002e-150

   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 77.1%

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

   if 7.40000000000000002e-150 < m

   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 38.7%

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

   4. Taylor expanded in v around 0 38.7%

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

      1. +-commutative 38.7%

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

      2. associate-*r* 38.7%

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

      3. neg-mul-1 38.7%

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

      4. distribute-rgt-out 38.7%

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

   6. Simplified 38.7%

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

   7. Taylor expanded in v around 0 29.4%

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

   8. Taylor expanded in m around 0 66.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.4 \cdot 10^{-150}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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* 99.8%

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

   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 99.8%

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

6. Taylor expanded in v around 0 100.0%

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

7. Taylor expanded in m around 0 81.8%

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

8. Final simplification 81.8%

   \[\leadsto -1 + \frac{m}{v} \]
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 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* 99.8%

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

   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 27.9%

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

   1. neg-mul-1 27.9%

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

   2. sub-neg 27.9%

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

   3. +-commutative 27.9%

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

   4. distribute-neg-in 27.9%

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

   5. remove-double-neg 27.9%

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

   6. metadata-eval 27.9%

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

7. Simplified 27.9%

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

8. Final simplification 27.9%

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

Alternative 11: 25.0% 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 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* 99.8%

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

   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 25.6%

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

6. Final simplification 25.6%

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

Reproduce

herbie shell --seed 2024066 
(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)))