a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.7%

Time: 8.0s

Alternatives: 11

Speedup: 1.0×

• 42.3% 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: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.4e-18) (fma m (/ m v) (- m)) (* (- 1.0 m) (/ (* m m) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.4e-18) {
		tmp = fma(m, (m / v), -m);
	} else {
		tmp = (1.0 - m) * ((m * m) / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.4e-18)
		tmp = fma(m, Float64(m / v), Float64(-m));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m * m) / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.4e-18], N[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.40000000000000006e-18

   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

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

      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. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. 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. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   6. 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. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.40000000000000006e-18 < 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

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

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

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. rgt-mult-inverse N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-*r/ N/A

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

      13. *-rgt-identity N/A

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

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

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

      15. div-sub N/A

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

      16. associate-/l* N/A

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

      17. lower-/.f64 N/A

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

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

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

      19. *-lft-identity N/A

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

      20. unpow2 N/A

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

      21. lower--.f64 N/A

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

      22. unpow2 N/A

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

      23. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

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

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

      2. lift-neg.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. sub-neg N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.8

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-*.f64 N/A

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

      18. lift-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m \cdot m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right)\\ t_1 := m \cdot \frac{m}{v}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+47}:\\ \;\;\;\;-t\_1\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (+ (/ (* m (- 1.0 m)) v) -1.0))) (t_1 (* m (/ m v))))
   (if (<= t_0 -4e+47) (- t_1) (if (<= t_0 -2e-308) (- m) t_1))))

C

double code(double m, double v) {
	double t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	double t_1 = m * (m / v);
	double tmp;
	if (t_0 <= -4e+47) {
		tmp = -t_1;
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = m * (((m * (1.0d0 - m)) / v) + (-1.0d0))
    t_1 = m * (m / v)
    if (t_0 <= (-4d+47)) then
        tmp = -t_1
    else if (t_0 <= (-2d-308)) then
        tmp = -m
    else
        tmp = t_1
    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 t_1 = m * (m / v);
	double tmp;
	if (t_0 <= -4e+47) {
		tmp = -t_1;
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0)
	t_1 = m * (m / v)
	tmp = 0
	if t_0 <= -4e+47:
		tmp = -t_1
	elif t_0 <= -2e-308:
		tmp = -m
	else:
		tmp = t_1
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0))
	t_1 = Float64(m * Float64(m / v))
	tmp = 0.0
	if (t_0 <= -4e+47)
		tmp = Float64(-t_1);
	elseif (t_0 <= -2e-308)
		tmp = Float64(-m);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	t_1 = m * (m / v);
	tmp = 0.0;
	if (t_0 <= -4e+47)
		tmp = -t_1;
	elseif (t_0 <= -2e-308)
		tmp = -m;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+47], (-t$95$1), If[LessEqual[t$95$0, -2e-308], (-m), t$95$1]]]]

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) < -4.0000000000000002e47

   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. cube-mult N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative 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%

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

   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. Applied rewrites 0.1%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 0.1

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

   10. Applied rewrites 0.1%

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

       1. frac-2neg N/A

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

       2. lift-neg.f64 N/A

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

       3. distribute-frac-neg N/A

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

       4. div-inv N/A

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

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

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

       6. neg-sub0 N/A

          \[\leadsto \left(\color{blue}{\left(0 - m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       7. flip3-- N/A

          \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {m}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       8. metadata-eval N/A

          \[\leadsto \left(\frac{\color{blue}{0} - {m}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       9. neg-sub0 N/A

          \[\leadsto \left(\frac{\color{blue}{\mathsf{neg}\left({m}^{3}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       10. cube-neg N/A

           \[\leadsto \left(\frac{\color{blue}{{\left(\mathsf{neg}\left(m\right)\right)}^{3}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       11. lift-neg.f64 N/A

           \[\leadsto \left(\frac{{\color{blue}{\left(\mathsf{neg}\left(m\right)\right)}}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       12. sqr-pow N/A

           \[\leadsto \left(\frac{\color{blue}{{\left(\mathsf{neg}\left(m\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(m\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       13. pow-prod-down N/A

           \[\leadsto \left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(m\right)\right) \cdot \left(\mathsf{neg}\left(m\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       14. lift-neg.f64 N/A

           \[\leadsto \left(\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(m\right)\right)} \cdot \left(\mathsf{neg}\left(m\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       15. lift-neg.f64 N/A

           \[\leadsto \left(\frac{{\left(\left(\mathsf{neg}\left(m\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       16. sqr-neg N/A

           \[\leadsto \left(\frac{{\color{blue}{\left(m \cdot m\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       17. pow-prod-down N/A

           \[\leadsto \left(\frac{\color{blue}{{m}^{\left(\frac{3}{2}\right)} \cdot {m}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       18. sqr-pow N/A

           \[\leadsto \left(\frac{\color{blue}{{m}^{3}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       19. associate-*l/ N/A

           \[\leadsto \color{blue}{\frac{{m}^{3} \cdot \frac{1}{\mathsf{neg}\left(v\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}} \cdot m \]

   12. Applied rewrites 78.3%

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

   if -4.0000000000000002e47 < (*.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 95.7

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

   5. Applied rewrites 95.7%

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

   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. cube-mult N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative 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 77.1%

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

   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 74.3

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

   8. Applied rewrites 74.3%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 92.9

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

   10. Applied rewrites 92.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -4 \cdot 10^{+47}:\\ \;\;\;\;-m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -4 \cdot 10^{+47}:\\ \;\;\;\;-m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -4e+47)
   (- (* m (/ m v)))
   (* m (+ -1.0 (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -4e+47) {
		tmp = -(m * (m / v));
	} else {
		tmp = m * (-1.0 + (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((m * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-4d+47)) then
        tmp = -(m * (m / v))
    else
        tmp = m * ((-1.0d0) + (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)) <= -4e+47) {
		tmp = -(m * (m / v));
	} else {
		tmp = m * (-1.0 + (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -4e+47:
		tmp = -(m * (m / v))
	else:
		tmp = m * (-1.0 + (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)) <= -4e+47)
		tmp = Float64(-Float64(m * Float64(m / v)));
	else
		tmp = Float64(m * Float64(-1.0 + 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)) <= -4e+47)
		tmp = -(m * (m / v));
	else
		tmp = m * (-1.0 + (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], -4e+47], (-N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $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) < -4.0000000000000002e47

   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. cube-mult N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative 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%

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

   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. Applied rewrites 0.1%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 0.1

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

   10. Applied rewrites 0.1%

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

       1. frac-2neg N/A

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

       2. lift-neg.f64 N/A

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

       3. distribute-frac-neg N/A

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

       4. div-inv N/A

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

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

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

       6. neg-sub0 N/A

          \[\leadsto \left(\color{blue}{\left(0 - m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       7. flip3-- N/A

          \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {m}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       8. metadata-eval N/A

          \[\leadsto \left(\frac{\color{blue}{0} - {m}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       9. neg-sub0 N/A

          \[\leadsto \left(\frac{\color{blue}{\mathsf{neg}\left({m}^{3}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       10. cube-neg N/A

           \[\leadsto \left(\frac{\color{blue}{{\left(\mathsf{neg}\left(m\right)\right)}^{3}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       11. lift-neg.f64 N/A

           \[\leadsto \left(\frac{{\color{blue}{\left(\mathsf{neg}\left(m\right)\right)}}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       12. sqr-pow N/A

           \[\leadsto \left(\frac{\color{blue}{{\left(\mathsf{neg}\left(m\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(m\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       13. pow-prod-down N/A

           \[\leadsto \left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(m\right)\right) \cdot \left(\mathsf{neg}\left(m\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       14. lift-neg.f64 N/A

           \[\leadsto \left(\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(m\right)\right)} \cdot \left(\mathsf{neg}\left(m\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       15. lift-neg.f64 N/A

           \[\leadsto \left(\frac{{\left(\left(\mathsf{neg}\left(m\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       16. sqr-neg N/A

           \[\leadsto \left(\frac{{\color{blue}{\left(m \cdot m\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       17. pow-prod-down N/A

           \[\leadsto \left(\frac{\color{blue}{{m}^{\left(\frac{3}{2}\right)} \cdot {m}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       18. sqr-pow N/A

           \[\leadsto \left(\frac{\color{blue}{{m}^{3}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       19. associate-*l/ N/A

           \[\leadsto \color{blue}{\frac{{m}^{3} \cdot \frac{1}{\mathsf{neg}\left(v\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}} \cdot m \]

   12. Applied rewrites 78.3%

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

   if -4.0000000000000002e47 < (*.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

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

      1. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -4 \cdot 10^{+47}:\\ \;\;\;\;-m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.1% 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 38.7

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

   5. Applied rewrites 38.7%

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

   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. cube-mult N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative 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 77.1%

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

   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 74.3

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

   8. Applied rewrites 74.3%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 92.9

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

   10. Applied rewrites 92.9%

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

4. Final simplification 52.4%

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

Alternative 5: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 97.8% accurate, 0.9× speedup

Math

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

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(m * Float64(Float64(m * Float64(-m)) / v));
	end
	return tmp
end

Wolfram

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

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. 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. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   6. 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. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

   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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 97.6%

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

Alternative 7: 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(m * Float64(m * m)) / Float64(-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. 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. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. 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. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   6. 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. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

   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. *-commutative N/A

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

      7. lower-neg.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. lower-/.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 97.5%

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

Alternative 8: 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 9: 87.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0) (fma m (/ m v) (- 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 / 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(m * Float64(m / v)));
	end
	return tmp
end

Wolfram

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

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. 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. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} - 1\right)} \]
   6. 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. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

   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

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   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. cube-mult N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative 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%

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

   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. Applied rewrites 0.1%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 0.1

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

   10. Applied rewrites 0.1%

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

       1. frac-2neg N/A

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

       2. lift-neg.f64 N/A

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

       3. distribute-frac-neg N/A

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

       4. div-inv N/A

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

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

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

       6. neg-sub0 N/A

          \[\leadsto \left(\color{blue}{\left(0 - m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       7. flip3-- N/A

          \[\leadsto \left(\color{blue}{\frac{{0}^{3} - {m}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       8. metadata-eval N/A

          \[\leadsto \left(\frac{\color{blue}{0} - {m}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       9. neg-sub0 N/A

          \[\leadsto \left(\frac{\color{blue}{\mathsf{neg}\left({m}^{3}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       10. cube-neg N/A

           \[\leadsto \left(\frac{\color{blue}{{\left(\mathsf{neg}\left(m\right)\right)}^{3}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       11. lift-neg.f64 N/A

           \[\leadsto \left(\frac{{\color{blue}{\left(\mathsf{neg}\left(m\right)\right)}}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       12. sqr-pow N/A

           \[\leadsto \left(\frac{\color{blue}{{\left(\mathsf{neg}\left(m\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(m\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       13. pow-prod-down N/A

           \[\leadsto \left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(m\right)\right) \cdot \left(\mathsf{neg}\left(m\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       14. lift-neg.f64 N/A

           \[\leadsto \left(\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(m\right)\right)} \cdot \left(\mathsf{neg}\left(m\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       15. lift-neg.f64 N/A

           \[\leadsto \left(\frac{{\left(\left(\mathsf{neg}\left(m\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       16. sqr-neg N/A

           \[\leadsto \left(\frac{{\color{blue}{\left(m \cdot m\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       17. pow-prod-down N/A

           \[\leadsto \left(\frac{\color{blue}{{m}^{\left(\frac{3}{2}\right)} \cdot {m}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       18. sqr-pow N/A

           \[\leadsto \left(\frac{\color{blue}{{m}^{3}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)} \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right) \cdot m \]

       19. associate-*l/ N/A

           \[\leadsto \color{blue}{\frac{{m}^{3} \cdot \frac{1}{\mathsf{neg}\left(v\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}} \cdot m \]

   12. Applied rewrites 78.3%

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

4. Final simplification 88.9%

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

Alternative 10: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   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. sub-neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 11: 26.8% 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 29.4

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

5. Applied rewrites 29.4%

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

Reproduce

herbie shell --seed 2024214 
(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))