b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.9s

Alternatives: 12

Speedup: N/A×

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

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. distribute-lft-in 99.8%

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

   2. *-commutative 99.8%

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

   3. neg-mul-1 99.8%

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

   4. associate-*r/ 99.9%

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

   5. div-inv 99.8%

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

   6. associate-*r* 99.8%

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

   7. fma-define 99.8%

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

6. Applied egg-rr 99.8%

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

   1. fmm-undef 99.8%

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

8. Simplified 99.8%

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

   1. associate-*l* 99.8%

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

   2. div-inv 99.9%

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

   3. associate-*l/ 99.9%

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

   4. associate-/r/ 99.9%

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

   5. associate-*r/ 99.9%

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

   6. *-commutative 99.9%

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

10. Applied egg-rr 99.9%

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

11. Taylor expanded in m around 0 99.9%

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

    1. expm1-log1p-u 50.7%

       \[\leadsto \frac{m \cdot \left(1 - m\right)}{\frac{v}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}}} - 1 \]

13. Applied egg-rr 50.7%

    \[\leadsto \frac{m \cdot \left(1 - m\right)}{\frac{v}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)}}} - 1 \]
14. Step-by-step derivation

    1. expm1-undefine 50.7%

       \[\leadsto \frac{m \cdot \left(1 - m\right)}{\frac{v}{\color{blue}{e^{\mathsf{log1p}\left(1 - m\right)} - 1}}} - 1 \]

    2. sub-neg 50.7%

       \[\leadsto \frac{m \cdot \left(1 - m\right)}{\frac{v}{\color{blue}{e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)}}} - 1 \]

    3. log1p-undefine 50.7%

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

    4. rem-exp-log 99.9%

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

    5. associate-+r- 99.9%

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

    6. metadata-eval 99.9%

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

    7. metadata-eval 99.9%

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

15. Simplified 99.9%

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

16. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 98.0%

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

   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 0 99.9%

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

      1. +-commutative 99.9%

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

      2. distribute-rgt-in 55.4%

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

      3. associate-/r/ 55.4%

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

      4. unpow-1 55.4%

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

      5. neg-mul-1 55.4%

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

      6. distribute-lft-neg-out 55.4%

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

      7. associate-/r/ 55.5%

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

      8. sub-neg 55.5%

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

      9. unpow-1 55.5%

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

      10. div-sub 99.9%

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

      11. associate-/r/ 99.9%

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

      12. associate-*l/ 99.9%

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

      13. associate-*r/ 99.9%

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

      14. *-commutative 99.9%

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

      15. associate-/r/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf 98.5%

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

      1. neg-mul-1 98.5%

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

   10. Simplified 98.5%

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

4. Final simplification 98.2%

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

Alternative 3: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 98.0%

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

   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 98.5%

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

      1. neg-mul-1 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 98.2%

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

Alternative 4: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(1.0 + N[(m / N[(v / m), $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 98.0%

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

   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. 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%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around inf 98.5%

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

      1. neg-mul-1 98.5%

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

   7. Simplified 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-/r/ 98.5%

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

   9. Applied egg-rr 98.5%

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

   10. Taylor expanded in m around inf 98.4%

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

       1. associate-*r/ 98.4%

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

       2. neg-mul-1 98.4%

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

   12. Simplified 98.4%

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

4. Final simplification 98.2%

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

Alternative 5: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(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) * (-1.0 + (m / v));
	else
		tmp = 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[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * 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 98.0%

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

   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. 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%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around inf 98.5%

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

      1. neg-mul-1 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in m around inf 98.4%

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

      1. neg-mul-1 98.5%

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

   10. Simplified 98.4%

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

4. Final simplification 98.2%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $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. Taylor expanded in m around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. distribute-rgt-in 77.9%

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

   3. associate-/r/ 78.0%

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

   4. unpow-1 78.0%

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

   5. neg-mul-1 78.0%

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

   6. distribute-lft-neg-out 78.0%

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

   7. associate-/r/ 78.0%

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

   8. sub-neg 78.0%

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

   9. unpow-1 78.0%

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

   10. div-sub 99.8%

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

   11. associate-/r/ 99.8%

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

   12. associate-*l/ 99.9%

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

   13. associate-*r/ 99.9%

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

   14. *-commutative 99.9%

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

   15. associate-/r/ 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 62.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-167}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.6e-167) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-167) {
		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 <= 1.6d-167) 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 <= 1.6e-167) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6e-167:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6e-167)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6e-167)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6e-167], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001e-167

   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 72.8%

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

   if 1.6000000000000001e-167 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-*r/ 99.9%

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

      5. div-inv 99.8%

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

      6. associate-*r* 99.8%

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

      7. fma-define 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. fmm-undef 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in m around 0 65.9%

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

   10. Taylor expanded in m around 0 65.9%

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

   11. Taylor expanded in m around inf 54.7%

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

Alternative 9: 26.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-58}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.5e-58) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.5e-58) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.5d-58) then
        tmp = -1.0d0
    else
        tmp = m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.5e-58) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.5e-58:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.5e-58)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.5e-58)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.5e-58], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 1.50000000000000004e-58

   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 58.6%

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

   if 1.50000000000000004e-58 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around inf 5.0%

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

      1. neg-mul-1 5.0%

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

      2. neg-sub0 5.0%

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

      3. associate--r- 5.0%

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

      4. metadata-eval 5.0%

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

   7. Simplified 5.0%

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

   8. Taylor expanded in m around inf 5.6%

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

Alternative 10: 75.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0 49.8%

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

4. Taylor expanded in m around 0 73.7%

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

5. Final simplification 73.7%

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

Alternative 11: 26.8% 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 26.8%

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

   1. neg-mul-1 26.8%

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

   2. neg-sub0 26.8%

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

   3. associate--r- 26.8%

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

   4. metadata-eval 26.8%

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

7. Simplified 26.8%

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

8. Final simplification 26.8%

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

Alternative 12: 24.4% 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 24.3%

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

Reproduce

herbie shell --seed 2024172 
(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)))