a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 6.4s

Alternatives: 12

Speedup: 1.1×

• 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;m \cdot \frac{m}{-v}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (+ (/ (* m (- 1.0 m)) v) -1.0))))
   (if (<= t_0 -2e+14)
     (* m (/ m (- v)))
     (if (<= t_0 -2e-308) (- m) (* m (/ m v))))))

C

double code(double m, double v) {
	double t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	double tmp;
	if (t_0 <= -2e+14) {
		tmp = m * (m / -v);
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (((m * (1.0d0 - m)) / v) + (-1.0d0))
    if (t_0 <= (-2d+14)) then
        tmp = m * (m / -v)
    else if (t_0 <= (-2d-308)) then
        tmp = -m
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	double tmp;
	if (t_0 <= -2e+14) {
		tmp = m * (m / -v);
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0)
	tmp = 0
	if t_0 <= -2e+14:
		tmp = m * (m / -v)
	elif t_0 <= -2e-308:
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0))
	tmp = 0.0
	if (t_0 <= -2e+14)
		tmp = Float64(m * Float64(m / Float64(-v)));
	elseif (t_0 <= -2e-308)
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	tmp = 0.0;
	if (t_0 <= -2e+14)
		tmp = m * (m / -v);
	elseif (t_0 <= -2e-308)
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+14], N[(m * N[(m / (-v)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-308], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -2e14

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{m \cdot v} \cdot {m}^{3} - \frac{1}{v} \cdot {m}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{m \cdot v}} - \frac{1}{v} \cdot {m}^{3} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. rgt-mult-inverse N/A

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

      9. *-commutative N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

      15. distribute-rgt-out-- N/A

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

      16. div-sub N/A

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

      17. associate-/l* N/A

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

      18. lower-/.f64 N/A

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

   5. Simplified 99.7%

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

   6. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 0.1

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

   8. Simplified 0.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 0.1

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

   10. Applied egg-rr 0.1%

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

   12. Applied egg-rr 73.2%

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

   if -2e14 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(m\right)} \]

      2. lower-neg.f64 95.6

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

   5. Simplified 95.6%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.6%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{m \cdot v} \cdot {m}^{3} - \frac{1}{v} \cdot {m}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{m \cdot v}} - \frac{1}{v} \cdot {m}^{3} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. rgt-mult-inverse N/A

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

      9. *-commutative N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

      15. distribute-rgt-out-- N/A

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

      16. div-sub N/A

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

      17. associate-/l* N/A

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

      18. lower-/.f64 N/A

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

   5. Simplified 76.5%

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

   6. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 72.9

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

   8. Simplified 72.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 86.8

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

   10. Applied egg-rr 86.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+14}:\\ \;\;\;\;m \cdot \frac{m}{-v}\\ \mathbf{elif}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+14}:\\ \;\;\;\;-\mathsf{fma}\left(m, \frac{m}{v}, m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e+14)
   (- (fma m (/ m v) m))
   (fma m (/ m v) (- m))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+14) {
		tmp = -fma(m, (m / v), m);
	} else {
		tmp = fma(m, (m / v), -m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e+14)
		tmp = Float64(-fma(m, Float64(m / v), m));
	else
		tmp = fma(m, Float64(m / v), Float64(-m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e+14], (-N[(m * N[(m / v), $MachinePrecision] + m), $MachinePrecision]), N[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -2e14

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 0.1

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

   5. Simplified 0.1%

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

   7. Applied egg-rr 73.2%

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

   if -2e14 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 97.3

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

   5. Simplified 97.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+14}:\\ \;\;\;\;-\mathsf{fma}\left(m, \frac{m}{v}, m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-\mathsf{fma}\left(m, \frac{m}{v}, m\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e-308)
   (- (fma m (/ m v) m))
   (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308) {
		tmp = -fma(m, (m / v), m);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e-308)
		tmp = Float64(-fma(m, Float64(m / v), m));
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e-308], (-N[(m * N[(m / v), $MachinePrecision] + m), $MachinePrecision]), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 34.2

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

   5. Simplified 34.2%

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

   7. Applied egg-rr 80.8%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.6%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{m \cdot v} \cdot {m}^{3} - \frac{1}{v} \cdot {m}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{m \cdot v}} - \frac{1}{v} \cdot {m}^{3} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. rgt-mult-inverse N/A

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

      9. *-commutative N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

      15. distribute-rgt-out-- N/A

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

      16. div-sub N/A

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

      17. associate-/l* N/A

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

      18. lower-/.f64 N/A

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

   5. Simplified 76.5%

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

   6. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 72.9

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

   8. Simplified 72.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 86.8

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

   10. Applied egg-rr 86.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-\mathsf{fma}\left(m, \frac{m}{v}, m\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e-308) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308) {
		tmp = -m;
	} else {
		tmp = 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 * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-2d-308)) then
        tmp = -m
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308:
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e-308)
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308)
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e-308], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(m\right)} \]

      2. lower-neg.f64 36.5

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

   5. Simplified 36.5%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.6%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{m \cdot v} \cdot {m}^{3} - \frac{1}{v} \cdot {m}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{m \cdot v}} - \frac{1}{v} \cdot {m}^{3} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. rgt-mult-inverse N/A

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

      9. *-commutative N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

      15. distribute-rgt-out-- N/A

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

      16. div-sub N/A

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

      17. associate-/l* N/A

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

      18. lower-/.f64 N/A

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

   5. Simplified 76.5%

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

   6. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 72.9

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

   8. Simplified 72.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 86.8

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

   10. Applied egg-rr 86.8%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.7 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \mathsf{fma}\left(-m, m, m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.7e-37) (fma m (/ m v) (- m)) (/ (* m (fma (- m) m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.7e-37) {
		tmp = fma(m, (m / v), -m);
	} else {
		tmp = (m * fma(-m, m, m)) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.7e-37)
		tmp = fma(m, Float64(m / v), Float64(-m));
	else
		tmp = Float64(Float64(m * fma(Float64(-m), m, m)) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.7e-37], N[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision], N[(N[(m * N[((-m) * m + m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.7e-37

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.8

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

   5. Simplified 99.8%

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

   if 3.7e-37 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{m \cdot v} \cdot {m}^{3} - \frac{1}{v} \cdot {m}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{m \cdot v}} - \frac{1}{v} \cdot {m}^{3} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. rgt-mult-inverse N/A

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

      9. *-commutative N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

      15. distribute-rgt-out-- N/A

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

      16. div-sub N/A

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

      17. associate-/l* N/A

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

      18. lower-/.f64 N/A

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

   5. Simplified 99.0%

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

      1. cancel-sign-sub-inv N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 99.1

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

   7. Applied egg-rr 99.1%

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

Alternative 7: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.7 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.7e-37) (fma m (/ m v) (- m)) (/ (* m (- m (* m m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.7e-37) {
		tmp = fma(m, (m / v), -m);
	} else {
		tmp = (m * (m - (m * m))) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.7e-37)
		tmp = fma(m, Float64(m / v), Float64(-m));
	else
		tmp = Float64(Float64(m * Float64(m - Float64(m * m))) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.7e-37], N[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision], N[(N[(m * N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.7e-37

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.8

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

   5. Simplified 99.8%

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

   if 3.7e-37 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{m \cdot v} \cdot {m}^{3} - \frac{1}{v} \cdot {m}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{m \cdot v}} - \frac{1}{v} \cdot {m}^{3} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. rgt-mult-inverse N/A

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

      9. *-commutative N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

      15. distribute-rgt-out-- N/A

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

      16. div-sub N/A

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

      17. associate-/l* N/A

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

      18. lower-/.f64 N/A

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

   5. Simplified 99.0%

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

Alternative 8: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.32 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m - m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.32e-37) (fma m (/ m v) (- m)) (* (/ m v) (- m (* m m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.32e-37) {
		tmp = fma(m, (m / v), -m);
	} else {
		tmp = (m / v) * (m - (m * m));
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.32e-37)
		tmp = fma(m, Float64(m / v), Float64(-m));
	else
		tmp = Float64(Float64(m / v) * Float64(m - Float64(m * m)));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.32e-37], N[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.3200000000000001e-37

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.8

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

   5. Simplified 99.8%

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

   if 1.3200000000000001e-37 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1}{m \cdot v} \cdot {m}^{3} - \frac{1}{v} \cdot {m}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{m \cdot v}} - \frac{1}{v} \cdot {m}^{3} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. rgt-mult-inverse N/A

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

      9. *-commutative N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l/ N/A

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

      14. *-lft-identity N/A

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

      15. distribute-rgt-out-- N/A

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

      16. div-sub N/A

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

      17. associate-/l* N/A

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

      18. lower-/.f64 N/A

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

   5. Simplified 99.0%

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

      1. cancel-sign-sub-inv N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 99.1

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

   7. Applied egg-rr 99.1%

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

      1. lift-neg.f64 N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. lift--.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft1-in N/A

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

      16. lift-fma.f64 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. div-inv N/A

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

   9. Applied egg-rr 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.32 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m - m \cdot m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 97.3

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

   5. Simplified 97.3%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {m}^{3}}{v}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{-1}{v} \cdot {m}^{3}} \]

      3. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-neg.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{m}^{3}}}{v}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{m}^{3}}{v}}\right) \]

      10. cube-mult N/A

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

      11. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{m \cdot \color{blue}{{m}^{2}}}{v}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{m \cdot {m}^{2}}}{v}\right) \]

      13. unpow2 N/A

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

      14. lower-*.f64 98.2

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

   5. Simplified 98.2%

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

Alternative 10: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 99.9

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 11: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

5. Simplified 99.8%

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

Alternative 12: 27.6% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := (-m)

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(m\right)} \]

   2. lower-neg.f64 28.4

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

5. Simplified 28.4%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (m v)
  :name "a 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) m))