a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 96.7%

Time: 6.7s

Alternatives: 11

Speedup: 1.1×

• 41.7% 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: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1e-100) (fma (/ m v) m (- m)) (/ (* (- 1.0 m) (* m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1e-100) {
		tmp = fma((m / v), m, -m);
	} else {
		tmp = ((1.0 - m) * (m * m)) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1e-100)
		tmp = fma(Float64(m / v), m, Float64(-m));
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(m * m)) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1e-100], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1e-100

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   8. 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-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 1e-100 < 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 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. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   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 5.6

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

   5. Applied rewrites 5.6%

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

   7. Applied rewrites 64.0%

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

   if -inf.0 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.3

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

   5. Applied rewrites 74.3%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. rgt-mult-inverse N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 94.7%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 94.8%

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

4. Final simplification 74.4%

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

Alternative 3: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+91}:\\ \;\;\;\;\frac{m}{v} \cdot \left(\left(-m\right) \cdot m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* m (- 1.0 m)) v) 1.0) m) -5e+91)
   (* (/ m v) (* (- m) m))
   (fma (/ m v) m (- m))))

C

double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -5e+91) {
		tmp = (m / v) * (-m * m);
	} else {
		tmp = fma((m / v), m, -m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m) <= -5e+91)
		tmp = Float64(Float64(m / v) * Float64(Float64(-m) * m));
	else
		tmp = fma(Float64(m / v), m, Float64(-m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -5e+91], N[(N[(m / v), $MachinePrecision] * N[((-m) * m), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * m + (-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) < -5.0000000000000002e91

   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. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in m around inf

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

   10. Applied rewrites 99.1%

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

   if -5.0000000000000002e91 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   8. 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-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.1

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

   9. Applied rewrites 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+91}:\\ \;\;\;\;\frac{m}{v} \cdot \left(\left(-m\right) \cdot m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\left(-m\right) \cdot m}{m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* m (- 1.0 m)) v) 1.0) m) -5e+91)
   (/ (* (- m) m) m)
   (fma (/ m v) m (- m))))

C

double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -5e+91) {
		tmp = (-m * m) / m;
	} else {
		tmp = fma((m / v), m, -m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m) <= -5e+91)
		tmp = Float64(Float64(Float64(-m) * m) / m);
	else
		tmp = fma(Float64(m / v), m, Float64(-m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -5e+91], N[(N[((-m) * m), $MachinePrecision] / m), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * m + (-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) < -5.0000000000000002e91

   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 5.4

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

   5. Applied rewrites 5.4%

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

   7. Applied rewrites 54.2%

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

   if -5.0000000000000002e91 < (*.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   8. 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-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.1

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

   9. Applied rewrites 99.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\left(-m\right) \cdot m}{m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -2e-308], (-m), N[(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) < -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 35.4

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

   5. Applied rewrites 35.4%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. rgt-mult-inverse N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 94.7%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 94.8%

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

4. Final simplification 48.9%

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

Alternative 6: 43.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -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 * (1.0d0 - m)) / v) - 1.0d0) * m) <= (-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 * (1.0 - m)) / v) - 1.0) * m) <= -2e-308) {
		tmp = -m;
	} else {
		tmp = (m * m) / v;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -2e-308], (-m), N[(N[(m * m), $MachinePrecision] / v), $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 35.4

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

   5. Applied rewrites 35.4%

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

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

   1. Initial program 99.5%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 73.4%

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

4. Final simplification 44.0%

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

Alternative 7: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1e-100) (fma (/ m v) m (- m)) (/ (* (* (- 1.0 m) m) m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1e-100) {
		tmp = fma((m / v), m, -m);
	} else {
		tmp = (((1.0 - m) * m) * m) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1e-100)
		tmp = fma(Float64(m / v), m, Float64(-m));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) * m) * m) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1e-100], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1e-100

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   8. 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-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 1e-100 < 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 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. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(\left(1 - m\right) \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1e-100) (fma (/ m v) m (- m)) (* (/ m v) (* (- 1.0 m) m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1e-100) {
		tmp = fma((m / v), m, -m);
	} else {
		tmp = (m / v) * ((1.0 - m) * m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1e-100)
		tmp = fma(Float64(m / v), m, Float64(-m));
	else
		tmp = Float64(Float64(m / v) * Float64(Float64(1.0 - m) * m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1e-100], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1e-100

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   8. 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-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 1e-100 < 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 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. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(\left(1 - m\right) \cdot m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. neg-mul-1 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 10: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 11: 27.5% 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 99.8%

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

3. Taylor expanded in m around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 27.9

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

5. Applied rewrites 27.9%

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

6. Final simplification 27.9%

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

Reproduce

herbie shell --seed 2024312 
(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))