b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.8%

Time: 8.5s

Alternatives: 11

Speedup: 1.0×

• 41.8% 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: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5.5e-16)
   (+ -1.0 (+ m (/ m v)))
   (* (- 1.0 m) (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.5e-16) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (1.0 - m) * (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 <= 5.5d-16) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (1.0d0 - m) * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5.5e-16) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.5e-16:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (1.0 - m) * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.5e-16)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.5e-16)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.5e-16], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.49999999999999964e-16

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(m \cdot \left(1 + \frac{1}{v}\right)\right)}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(1 \cdot m + \color{blue}{\frac{1}{v} \cdot m}\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \color{blue}{\frac{1}{v}} \cdot m\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{1 \cdot m}{\color{blue}{v}}\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m}{v}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

      10. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

   5. Simplified 100.0%

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

   if 5.49999999999999964e-16 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)}, \mathsf{\_.f64}\left(1, m\right)\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(m \cdot m\right) \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(m \cdot \frac{1}{m \cdot v}\right) - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(m \cdot \frac{\frac{1}{m}}{v}\right) - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{m \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      6. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - \left(m \cdot m\right) \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \left(m \cdot \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m \cdot 1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      11. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} - \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{m}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{m}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      18. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} - \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      19. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1 - m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \mathsf{/.f64}\left(\left(1 - m\right), v\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.43:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5.2e-134)
   -1.0
   (if (<= m 0.43) (* (- 1.0 m) (/ m v)) (* m (* m (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.2e-134) {
		tmp = -1.0;
	} else if (m <= 0.43) {
		tmp = (1.0 - m) * (m / v);
	} else {
		tmp = m * (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 <= 5.2d-134) then
        tmp = -1.0d0
    else if (m <= 0.43d0) then
        tmp = (1.0d0 - m) * (m / v)
    else
        tmp = m * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5.2e-134) {
		tmp = -1.0;
	} else if (m <= 0.43) {
		tmp = (1.0 - m) * (m / v);
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.2e-134:
		tmp = -1.0
	elif m <= 0.43:
		tmp = (1.0 - m) * (m / v)
	else:
		tmp = m * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.2e-134)
		tmp = -1.0;
	elseif (m <= 0.43)
		tmp = Float64(Float64(1.0 - m) * Float64(m / v));
	else
		tmp = Float64(m * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.2e-134)
		tmp = -1.0;
	elseif (m <= 0.43)
		tmp = (1.0 - m) * (m / v);
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.2e-134], -1.0, If[LessEqual[m, 0.43], N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 5.20000000000000045e-134

   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

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 73.1%

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

   if 5.20000000000000045e-134 < m < 0.429999999999999993

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)}, \mathsf{\_.f64}\left(1, m\right)\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(m \cdot m\right) \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(m \cdot \frac{1}{m \cdot v}\right) - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(m \cdot \frac{\frac{1}{m}}{v}\right) - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{m \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      6. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - \left(m \cdot m\right) \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \left(m \cdot \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m \cdot 1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      11. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} - \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{m}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{m}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      18. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} - \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      19. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1 - m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \mathsf{/.f64}\left(\left(1 - m\right), v\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   5. Simplified 75.7%

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

   6. Taylor expanded in m around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{m}{v}\right)}, \mathsf{\_.f64}\left(1, m\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, v\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

   8. Simplified 71.0%

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

   if 0.429999999999999993 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{m \cdot \left(m \cdot m\right)}{v} \]

      2. unpow2 N/A

         \[\leadsto \frac{m \cdot {m}^{2}}{v} \]

      3. associate-/l* N/A

         \[\leadsto m \cdot \color{blue}{\frac{{m}^{2}}{v}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(m, \color{blue}{\left(\frac{{m}^{2}}{v}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(m, \left(\frac{m \cdot m}{v}\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(m, \left(m \cdot \color{blue}{\frac{m}{v}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{*.f64}\left(m, \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

      8. /-lowering-/.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{*.f64}\left(m, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

   5. Simplified 98.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.43:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.4 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.4e-134) -1.0 (if (<= m 2.6) (/ m v) (* m (* m (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.4e-134) {
		tmp = -1.0;
	} else if (m <= 2.6) {
		tmp = m / v;
	} else {
		tmp = m * (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 <= 3.4d-134) then
        tmp = -1.0d0
    else if (m <= 2.6d0) then
        tmp = m / v
    else
        tmp = m * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.4e-134) {
		tmp = -1.0;
	} else if (m <= 2.6) {
		tmp = m / v;
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.4e-134:
		tmp = -1.0
	elif m <= 2.6:
		tmp = m / v
	else:
		tmp = m * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.4e-134)
		tmp = -1.0;
	elseif (m <= 2.6)
		tmp = Float64(m / v);
	else
		tmp = Float64(m * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.4e-134)
		tmp = -1.0;
	elseif (m <= 2.6)
		tmp = m / v;
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.4e-134], -1.0, If[LessEqual[m, 2.6], N[(m / v), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 3.39999999999999977e-134

   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

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 73.1%

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

   if 3.39999999999999977e-134 < m < 2.60000000000000009

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)}, \mathsf{\_.f64}\left(1, m\right)\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(m \cdot m\right) \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(m \cdot \frac{1}{m \cdot v}\right) - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(m \cdot \frac{\frac{1}{m}}{v}\right) - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{m \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      6. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - \left(m \cdot m\right) \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \left(m \cdot \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m \cdot 1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      11. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} - \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{m}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{m}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      18. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} - \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      19. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1 - m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \mathsf{/.f64}\left(\left(1 - m\right), v\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   5. Simplified 75.7%

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

   6. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{m}{v}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 70.8%

         \[\leadsto \mathsf{/.f64}\left(m, \color{blue}{v}\right) \]

   8. Simplified 70.8%

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

   if 2.60000000000000009 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{m \cdot \left(m \cdot m\right)}{v} \]

      2. unpow2 N/A

         \[\leadsto \frac{m \cdot {m}^{2}}{v} \]

      3. associate-/l* N/A

         \[\leadsto m \cdot \color{blue}{\frac{{m}^{2}}{v}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(m, \color{blue}{\left(\frac{{m}^{2}}{v}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(m, \left(\frac{m \cdot m}{v}\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(m, \left(m \cdot \color{blue}{\frac{m}{v}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{*.f64}\left(m, \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

      8. /-lowering-/.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{*.f64}\left(m, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

   5. Simplified 98.0%

      \[\leadsto \color{blue}{m \cdot \left(m \cdot \frac{m}{v}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(m + -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- 1.0 m) (+ -1.0 (/ m v))) (* m (* (/ m v) (+ m -2.0)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * ((m / v) * (m + -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (m / v))
    else
        tmp = m * ((m / v) * (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * ((m / v) * (m + -2.0));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = m * ((m / v) * (m + -2.0))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(Float64(m / v) * Float64(m + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	else
		tmp = m * ((m / v) * (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m / v), $MachinePrecision] * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{m}{v}\right)}, 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 97.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, v\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   5. Simplified 97.7%

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

   if 1.6000000000000001 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]

   5. Simplified 99.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(m + -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.45:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(m + -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.45) (+ -1.0 (+ m (/ m v))) (* m (* (/ m v) (+ m -2.0)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.45) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * ((m / v) * (m + -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.45d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = m * ((m / v) * (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.45) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * ((m / v) * (m + -2.0));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.45:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = m * ((m / v) * (m + -2.0))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.45)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(m * Float64(Float64(m / v) * Float64(m + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.45)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = m * ((m / v) * (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.45], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m / v), $MachinePrecision] * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.4500000000000002

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(m \cdot \left(1 + \frac{1}{v}\right)\right)}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(1 \cdot m + \color{blue}{\frac{1}{v} \cdot m}\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \color{blue}{\frac{1}{v}} \cdot m\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{1 \cdot m}{\color{blue}{v}}\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m}{v}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

      10. /-lowering-/.f64 97.6%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

   5. Simplified 97.6%

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

   if 2.4500000000000002 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]

   5. Simplified 99.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.45:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(m + -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 7: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.65:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.65) (+ -1.0 (+ m (/ m v))) (* m (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.65) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * (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 <= 2.65d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = m * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.65) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.65:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = m * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.65)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(m * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.65)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.65], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.64999999999999991

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(m \cdot \left(1 + \frac{1}{v}\right)\right)}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(1 \cdot m + \color{blue}{\frac{1}{v} \cdot m}\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \color{blue}{\frac{1}{v}} \cdot m\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{1 \cdot m}{\color{blue}{v}}\right)\right) \]

      8. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m}{v}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

      10. /-lowering-/.f64 97.6%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

   5. Simplified 97.6%

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

   if 2.64999999999999991 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{m \cdot \left(m \cdot m\right)}{v} \]

      2. unpow2 N/A

         \[\leadsto \frac{m \cdot {m}^{2}}{v} \]

      3. associate-/l* N/A

         \[\leadsto m \cdot \color{blue}{\frac{{m}^{2}}{v}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(m, \color{blue}{\left(\frac{{m}^{2}}{v}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(m, \left(\frac{m \cdot m}{v}\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(m, \left(m \cdot \color{blue}{\frac{m}{v}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{*.f64}\left(m, \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

      8. /-lowering-/.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{*.f64}\left(m, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

   5. Simplified 98.0%

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

Alternative 8: 62.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3.6e-134) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.6e-134) {
		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 <= 3.6d-134) 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 <= 3.6e-134) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.6e-134:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.6e-134)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.6e-134)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.6e-134], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.5999999999999999e-134

   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

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 73.1%

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

   if 3.5999999999999999e-134 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)}, \mathsf{\_.f64}\left(1, m\right)\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(m \cdot m\right) \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(m \cdot \frac{1}{m \cdot v}\right) - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(m \cdot \frac{\frac{1}{m}}{v}\right) - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{m \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      6. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - {m}^{2} \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - \left(m \cdot m\right) \cdot \frac{1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \left(m \cdot \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m \cdot 1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      11. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} - \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{m}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{m}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      18. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1}{v} - \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      19. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{1 - m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, \mathsf{/.f64}\left(\left(1 - m\right), v\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   5. Simplified 92.3%

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

   6. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{m}{v}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 59.1%

         \[\leadsto \mathsf{/.f64}\left(m, \color{blue}{v}\right) \]

   8. Simplified 59.1%

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

Alternative 9: 27.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.7 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.7e-16) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.7e-16) {
		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 <= 2.7d-16) 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 <= 2.7e-16) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.7e-16:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.7e-16)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.7e-16)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.7e-16], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 2.69999999999999999e-16

   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

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 52.1%

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

   if 2.69999999999999999e-16 < 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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(1 - m\right)\right) \]

      2. neg-sub0 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

         \[\leadsto -1 + m \]

      5. +-commutative N/A

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

      6. +-lowering-+.f64 5.1%

         \[\leadsto \mathsf{+.f64}\left(m, \color{blue}{-1}\right) \]

   7. Simplified 5.1%

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

   8. Taylor expanded in m around inf

      \[\leadsto \color{blue}{m} \]
   9. Step-by-step derivation

   10. Simplified 5.4%

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

Alternative 10: 27.5% 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. Add Preprocessing

3. Taylor expanded in v around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\left(1 - m\right)\right) \]

   2. neg-sub0 N/A

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

   3. associate--r- N/A

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

   4. metadata-eval N/A

      \[\leadsto -1 + m \]

   5. +-lowering-+.f64 26.8%

      \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{m}\right) \]

5. Simplified 26.8%

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

6. Final simplification 26.8%

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

Alternative 11: 25.1% 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. Add Preprocessing

3. Taylor expanded in m around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Simplified 24.4%

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

Reproduce

herbie shell --seed 2024191 
(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)))