a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.7%

Time: 6.9s

Alternatives: 10

Speedup: N/A×

• 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.6 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1 - m}{v} \cdot m\right) \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.6e-14) (fma (/ m v) m (- m)) (* (* (/ (- 1.0 m) v) m) m)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.6e-14) {
		tmp = fma((m / v), m, -m);
	} else {
		tmp = (((1.0 - m) / v) * m) * m;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.6e-14)
		tmp = fma(Float64(m / v), m, Float64(-m));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) / v) * m) * m);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.6e-14], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision] * m), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.5999999999999996e-14

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 6.5999999999999996e-14 < m

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. 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. clear-num N/A

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

      5. frac-2neg N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. sqr-neg N/A

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

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

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

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

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

      21. lower-*.f64 N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. neg-sub0 N/A

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

      6. associate-+l- N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft1-in N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. distribute-lft-in N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. div-sub N/A

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

      21. lower-/.f64 N/A

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

      22. lower--.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(\left(-m\right) \cdot m\right) \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* (- 1.0 m) m) v) 1.0) m) -5e+65)
   (/ (* (* (- m) m) m) v)
   (fma (/ m v) m (- m))))

C

double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+65) {
		tmp = ((-m * m) * m) / v;
	} else {
		tmp = fma((m / v), m, -m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(1.0 - m) * m) / v) - 1.0) * m) <= -5e+65)
		tmp = Float64(Float64(Float64(Float64(-m) * m) * m) / v);
	else
		tmp = fma(Float64(m / v), m, Float64(-m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -5e+65], N[(N[(N[((-m) * m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -4.99999999999999973e65

   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 v around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 99.0%

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

   if -4.99999999999999973e65 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(\left(-m\right) \cdot m\right) \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+65}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* (- 1.0 m) m) v) 1.0) m) -5e+65)
   (- m)
   (fma (/ m v) m (- m))))

C

double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+65) {
		tmp = -m;
	} else {
		tmp = fma((m / v), m, -m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(1.0 - m) * m) / v) - 1.0) * m) <= -5e+65)
		tmp = Float64(-m);
	else
		tmp = fma(Float64(m / v), m, Float64(-m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -5e+65], (-m), N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -4.99999999999999973e65

   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 v around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.5

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

   5. Applied rewrites 5.5%

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

   if -4.99999999999999973e65 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+65}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -4 \cdot 10^{-305}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* (- 1.0 m) m) v) 1.0) m) -4e-305) (- m) (* (/ m v) m)))

C

double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -4e-305) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((((((1.0d0 - m) * m) / v) - 1.0d0) * m) <= (-4d-305)) then
        tmp = -m
    else
        tmp = (m / v) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -4e-305) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (((((1.0 - m) * m) / v) - 1.0) * m) <= -4e-305:
		tmp = -m
	else:
		tmp = (m / v) * m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(1.0 - m) * m) / v) - 1.0) * m) <= -4e-305)
		tmp = Float64(-m);
	else
		tmp = Float64(Float64(m / v) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -4e-305)
		tmp = -m;
	else
		tmp = (m / v) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -4e-305], (-m), N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -3.99999999999999999e-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 v around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 31.7

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

   5. Applied rewrites 31.7%

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

   if -3.99999999999999999e-305 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.3%

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

   3. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 92.1%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -4 \cdot 10^{-305}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.6 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} \cdot m\right) \cdot \left(1 - m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.6e-14) (fma (/ m v) m (- m)) (* (* (/ m v) m) (- 1.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.6e-14) {
		tmp = fma((m / v), m, -m);
	} else {
		tmp = ((m / v) * m) * (1.0 - m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.6e-14)
		tmp = fma(Float64(m / v), m, Float64(-m));
	else
		tmp = Float64(Float64(Float64(m / v) * m) * Float64(1.0 - m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.6e-14], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision], N[(N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.5999999999999996e-14

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 6.5999999999999996e-14 < m

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 15.3%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 6: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.6 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.6e-14) (fma (/ m v) m (- m)) (/ (* (* (- 1.0 m) m) m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.6e-14) {
		tmp = fma((m / v), m, -m);
	} else {
		tmp = (((1.0 - m) * m) * m) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.6e-14)
		tmp = fma(Float64(m / v), m, Float64(-m));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) * m) * m) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.6e-14], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.5999999999999996e-14

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 6.5999999999999996e-14 < 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 v around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 7: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

   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. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 8: 74.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

   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 v around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.5

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

   5. Applied rewrites 5.5%

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

   7. Applied rewrites 53.8%

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

4. Final simplification 73.4%

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

Alternative 9: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[m_, v_] := N[(N[(N[(m - N[(m * m + v), $MachinePrecision]), $MachinePrecision] / 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. 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. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. associate-*r/ N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f64 55.7

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

5. Applied rewrites 55.7%

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

6. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. associate-+l- N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

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

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

   11. mul-1-neg N/A

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

   12. associate-+l- N/A

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

   13. neg-sub0 N/A

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

   14. mul-1-neg N/A

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

   15. lower-/.f64 N/A

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

8. Applied rewrites 99.8%

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

Alternative 10: 27.1% 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 v around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 24.5

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

5. Applied rewrites 24.5%

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

Reproduce

herbie shell --seed 2024284 
(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))