a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.7%

Time: 7.7s

Alternatives: 11

Speedup: 1.0×

• 41.5% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.5e-16) (- (/ m (/ v m)) m) (* m (* (/ m v) (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.5e-16) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * ((m / v) * (1.0 - m));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.5d-16) then
        tmp = (m / (v / m)) - m
    else
        tmp = m * ((m / v) * (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.5e-16) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * ((m / v) * (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.5e-16:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * ((m / v) * (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.5e-16)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m * Float64(Float64(m / v) * Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.5e-16)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * ((m / v) * (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.5e-16], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.49999999999999997e-16

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0

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

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

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   7. Applied rewrites 99.8%

      \[\leadsto \frac{m}{\frac{v}{m}} - m \]

   if 1.49999999999999997e-16 < 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. rgt-mult-inverse N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

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

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

      16. *-lft-identity N/A

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

      17. unpow2 N/A

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

      18. lower--.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(-m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ -1.0 (/ (* m (- 1.0 m)) v))) -2e+74)
   (* (/ m v) (* m (- m)))
   (fma m (/ m v) (- m))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -2e+74) {
		tmp = (m / v) * (m * -m);
	} else {
		tmp = fma(m, (m / v), -m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v))) <= -2e+74)
		tmp = Float64(Float64(m / v) * Float64(m * Float64(-m)));
	else
		tmp = fma(m, Float64(m / v), Float64(-m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(-1.0 + N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+74], N[(N[(m / v), $MachinePrecision] * N[(m * (-m)), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999999e74

   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. lower-/.f64 N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 98.7

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

   7. Applied rewrites 98.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(-m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{-305}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ -1.0 (/ (* m (- 1.0 m)) v))) -5e-305) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e-305) {
		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 * (1.0d0 - m)) / v))) <= (-5d-305)) 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 * (1.0 - m)) / v))) <= -5e-305) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e-305:
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v))) <= -5e-305)
		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 * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e-305)
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(-1.0 + N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-305], (-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) < -4.99999999999999985e-305

   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 34.2

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

   5. Applied rewrites 34.2%

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

   if -4.99999999999999985e-305 < (*.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. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. rgt-mult-inverse N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

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

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

      16. *-lft-identity N/A

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

      17. unpow2 N/A

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

      18. lower--.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 92.2%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{-305}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{-305}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ -1.0 (/ (* m (- 1.0 m)) v))) -5e-305) (- m) (/ (* m m) v)))

C

double code(double m, double v) {
	double tmp;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e-305) {
		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 * (1.0d0 - m)) / v))) <= (-5d-305)) 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 * (1.0 - m)) / v))) <= -5e-305) {
		tmp = -m;
	} else {
		tmp = (m * m) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e-305:
		tmp = -m
	else:
		tmp = (m * m) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v))) <= -5e-305)
		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 * (1.0 - m)) / v))) <= -5e-305)
		tmp = -m;
	else
		tmp = (m * m) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(-1.0 + N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-305], (-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) < -4.99999999999999985e-305

   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 34.2

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

   5. Applied rewrites 34.2%

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

   if -4.99999999999999985e-305 < (*.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 0

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

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

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 67.8%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{-305}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.5e-16) (fma m (/ m v) (- m)) (* m (* (/ m v) (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.5e-16) {
		tmp = fma(m, (m / v), -m);
	} else {
		tmp = m * ((m / v) * (1.0 - m));
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.5e-16)
		tmp = fma(m, Float64(m / v), Float64(-m));
	else
		tmp = Float64(m * Float64(Float64(m / v) * Float64(1.0 - m)));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.5e-16], N[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision], N[(m * N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.49999999999999997e-16

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.49999999999999997e-16 < 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. rgt-mult-inverse N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

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

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

      16. *-lft-identity N/A

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

      17. unpow2 N/A

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

      18. lower--.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. *-rgt-identity N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift--.f64 N/A

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

   17. sub-neg N/A

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

   18. +-commutative N/A

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

   19. distribute-lft-in N/A

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

   20. *-rgt-identity N/A

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

   21. lower-fma.f64 N/A

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

   22. lower-neg.f64 N/A

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

   23. lower-/.f64 N/A

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

   24. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 7: 74.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 98.7

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

   7. Applied rewrites 98.7%

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

   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 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 5.8

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 52.1%

      \[\leadsto \frac{m \cdot \left(-m\right)}{\color{blue}{0 + m}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

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

Alternative 8: 51.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 98.7

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

   7. Applied rewrites 98.7%

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

   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 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 5.8

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

   5. Applied rewrites 5.8%

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

Alternative 9: 51.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0

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

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

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 98.7%

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

   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 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 5.8

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

   5. Applied rewrites 5.8%

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

4. Final simplification 50.1%

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

Alternative 10: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   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. Final simplification 99.8%

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

Alternative 11: 26.9% 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 27.0

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

5. Applied rewrites 27.0%

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

Reproduce

herbie shell --seed 2024234 
(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))