a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 7.1s

Alternatives: 10

Speedup: 1.1×

• 41.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 - m, m \cdot \frac{m}{v}, 0 - m\right) \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (fma (- 1.0 m) (* m (/ m v)) (- 0.0 m)))

C

double code(double m, double v) {
	return fma((1.0 - m), (m * (m / v)), (0.0 - m));
}

Julia

function code(m, v)
	return fma(Float64(1.0 - m), Float64(m * Float64(m / v)), Float64(0.0 - m))
end

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] + N[(0.0 - m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. neg-mul-1 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. --lowering--.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. /-lowering-/.f64 N/A

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

   13. neg-sub0 N/A

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

   14. --lowering--.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(1 - m, m \cdot \frac{m}{v}, 0 - m\right) \]
6. Add Preprocessing

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{0 - v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e+18)
   (/ (* m (* m m)) (- 0.0 v))
   (- (* m (/ m v)) m)))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18) {
		tmp = (m * (m * m)) / (0.0 - v);
	} else {
		tmp = (m * (m / v)) - m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((m * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-2d+18)) then
        tmp = (m * (m * m)) / (0.0d0 - v)
    else
        tmp = (m * (m / v)) - m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18) {
		tmp = (m * (m * m)) / (0.0 - v);
	} else {
		tmp = (m * (m / v)) - m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18:
		tmp = (m * (m * m)) / (0.0 - v)
	else:
		tmp = (m * (m / v)) - m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e+18)
		tmp = Float64(Float64(m * Float64(m * m)) / Float64(0.0 - v));
	else
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18)
		tmp = (m * (m * m)) / (0.0 - v);
	else
		tmp = (m * (m / v)) - m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e+18], N[(N[(m * N[(m * m), $MachinePrecision]), $MachinePrecision] / N[(0.0 - v), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m / v), $MachinePrecision]), $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) < -2e18

   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. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. mul-1-neg N/A

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

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

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

      6. +-rgt-identity N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

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

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

      13. mul-1-neg N/A

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

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{m}{v}, -1 \cdot {m}^{2}, 0\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{m}{v}}, -1 \cdot {m}^{2}, 0\right) \]

      16. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{m}{v}, \color{blue}{\mathsf{neg}\left({m}^{2}\right)}, 0\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{m}{v}, \color{blue}{0 - {m}^{2}}, 0\right) \]

      18. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{m}{v}, \color{blue}{0 - {m}^{2}}, 0\right) \]

      19. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(\frac{m}{v}, 0 - \color{blue}{\left({m}^{2} + 0\right)}, 0\right) \]

      20. unpow2 N/A

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

      21. accelerator-lowering-fma.f64 96.9

          \[\leadsto \mathsf{fma}\left(\frac{m}{v}, 0 - \color{blue}{\mathsf{fma}\left(m, m, 0\right)}, 0\right) \]

   5. Simplified 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{m}{v}, 0 - \mathsf{fma}\left(m, m, 0\right), 0\right)} \]

   7. Applied egg-rr 96.9%

      \[\leadsto \color{blue}{-\frac{m \cdot \mathsf{fma}\left(m, m, 0\right)}{v}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 96.9

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

   9. Applied egg-rr 96.9%

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

   if -2e18 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. neg-sub0 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{m}{v} \cdot m, 0 - m\right) \]
   6. Step-by-step derivation

   7. Simplified 97.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{m}{v} \cdot m, 0 - m\right) \]
   8. Step-by-step derivation

      1. associate-+r- N/A

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

      2. +-rgt-identity N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{m \cdot m + 0}}{v} - m \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{m \cdot m + 0}{v}} - m \]

      8. accelerator-lowering-fma.f64 84.2

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(m, m, 0\right)}}{v} - m \]

   9. Applied egg-rr 84.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(m, m, 0\right)}{v} - m} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

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

       2. associate-*l/ N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 97.1

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

   11. Applied egg-rr 97.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{0 - v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{0 - \mathsf{fma}\left(m, m, 0\right)}{m}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e+18)
   (/ (- 0.0 (fma m m 0.0)) m)
   (- (* m (/ m v)) m)))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18) {
		tmp = (0.0 - fma(m, m, 0.0)) / m;
	} else {
		tmp = (m * (m / v)) - m;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e+18)
		tmp = Float64(Float64(0.0 - fma(m, m, 0.0)) / m);
	else
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e+18], N[(N[(0.0 - N[(m * m + 0.0), $MachinePrecision]), $MachinePrecision] / m), $MachinePrecision], N[(N[(m * N[(m / v), $MachinePrecision]), $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) < -2e18

   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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - m} \]

      3. --lowering--.f64 5.4

         \[\leadsto \color{blue}{0 - m} \]

   5. Simplified 5.4%

      \[\leadsto \color{blue}{0 - m} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 5.4

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

   7. Applied egg-rr 5.4%

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

      1. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - m} \]

      2. flip-- N/A

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - m \cdot m}{0 + m}} \]

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. neg-sub0 N/A

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

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

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

      7. distribute-lft-neg-out N/A

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

      8. sqr-neg N/A

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

      9. +-lft-identity N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-rgt-identity N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(m, m, 0\right)}}{\mathsf{neg}\left(m\right)} \]

      13. neg-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(m, m, 0\right)}{\color{blue}{0 - m}} \]

      14. --lowering--.f64 48.2

          \[\leadsto \frac{\mathsf{fma}\left(m, m, 0\right)}{\color{blue}{0 - m}} \]

   9. Applied egg-rr 48.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(m, m, 0\right)}{0 - m}} \]

   if -2e18 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. neg-sub0 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{m}{v} \cdot m, 0 - m\right) \]
   6. Step-by-step derivation

   7. Simplified 97.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{m}{v} \cdot m, 0 - m\right) \]
   8. Step-by-step derivation

      1. associate-+r- N/A

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

      2. +-rgt-identity N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{m \cdot m + 0}}{v} - m \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{m \cdot m + 0}{v}} - m \]

      8. accelerator-lowering-fma.f64 84.2

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(m, m, 0\right)}}{v} - m \]

   9. Applied egg-rr 84.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(m, m, 0\right)}{v} - m} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

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

       2. associate-*l/ N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 97.1

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

   11. Applied egg-rr 97.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{0 - \mathsf{fma}\left(m, m, 0\right)}{m}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+18}:\\ \;\;\;\;0 - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e+18)
   (- 0.0 m)
   (- (* m (/ m v)) m)))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18) {
		tmp = 0.0 - m;
	} else {
		tmp = (m * (m / v)) - m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((m * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-2d+18)) then
        tmp = 0.0d0 - m
    else
        tmp = (m * (m / v)) - m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18) {
		tmp = 0.0 - m;
	} else {
		tmp = (m * (m / v)) - m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18:
		tmp = 0.0 - m
	else:
		tmp = (m * (m / v)) - m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e+18)
		tmp = Float64(0.0 - m);
	else
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18)
		tmp = 0.0 - m;
	else
		tmp = (m * (m / v)) - m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e+18], N[(0.0 - m), $MachinePrecision], N[(N[(m * N[(m / v), $MachinePrecision]), $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) < -2e18

   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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - m} \]

      3. --lowering--.f64 5.4

         \[\leadsto \color{blue}{0 - m} \]

   5. Simplified 5.4%

      \[\leadsto \color{blue}{0 - m} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 5.4

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

   7. Applied egg-rr 5.4%

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

   if -2e18 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. neg-sub0 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{m}{v} \cdot m, 0 - m\right) \]
   6. Step-by-step derivation

   7. Simplified 97.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{m}{v} \cdot m, 0 - m\right) \]
   8. Step-by-step derivation

      1. associate-+r- N/A

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

      2. +-rgt-identity N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{m \cdot m + 0}}{v} - m \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{m \cdot m + 0}{v}} - m \]

      8. accelerator-lowering-fma.f64 84.2

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(m, m, 0\right)}}{v} - m \]

   9. Applied egg-rr 84.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(m, m, 0\right)}{v} - m} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

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

       2. associate-*l/ N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 97.1

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

   11. Applied egg-rr 97.1%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+18}:\\ \;\;\;\;0 - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+18}:\\ \;\;\;\;0 - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e+18)
   (- 0.0 m)
   (* m (+ (/ m v) -1.0))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18) {
		tmp = 0.0 - m;
	} else {
		tmp = m * ((m / v) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((m * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-2d+18)) then
        tmp = 0.0d0 - m
    else
        tmp = m * ((m / v) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18) {
		tmp = 0.0 - m;
	} else {
		tmp = m * ((m / v) + -1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18:
		tmp = 0.0 - m
	else:
		tmp = m * ((m / v) + -1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e+18)
		tmp = Float64(0.0 - m);
	else
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e+18)
		tmp = 0.0 - m;
	else
		tmp = m * ((m / v) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e+18], N[(0.0 - m), $MachinePrecision], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $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) < -2e18

   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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - m} \]

      3. --lowering--.f64 5.4

         \[\leadsto \color{blue}{0 - m} \]

   5. Simplified 5.4%

      \[\leadsto \color{blue}{0 - m} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 5.4

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

   7. Applied egg-rr 5.4%

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

   if -2e18 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 97.1

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

   5. Simplified 97.1%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{+18}:\\ \;\;\;\;0 - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -1 \cdot 10^{-308}:\\ \;\;\;\;0 - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -1e-308)
   (- 0.0 m)
   (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -1e-308) {
		tmp = 0.0 - m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((m * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-1d-308)) then
        tmp = 0.0d0 - m
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -1e-308) {
		tmp = 0.0 - m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -1e-308:
		tmp = 0.0 - m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -1e-308)
		tmp = Float64(0.0 - m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -1e-308)
		tmp = 0.0 - m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -1e-308], N[(0.0 - m), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -9.9999999999999991e-309

   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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - m} \]

      3. --lowering--.f64 39.7

         \[\leadsto \color{blue}{0 - m} \]

   5. Simplified 39.7%

      \[\leadsto \color{blue}{0 - m} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 39.7

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

   7. Applied egg-rr 39.7%

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

   if -9.9999999999999991e-309 < (*.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. Step-by-step derivation

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. neg-sub0 N/A

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

      14. --lowering--.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in m around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{m}{v} \cdot m, 0 - m\right) \]
   6. Step-by-step derivation

   7. Simplified 93.7%

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

   8. Taylor expanded in m around inf

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-rgt-identity N/A

         \[\leadsto \color{blue}{{m}^{2} \cdot \frac{1}{v} + 0} \]

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(m, \frac{\color{blue}{m}}{v}, 0\right) \]

      10. /-lowering-/.f64 88.0

          \[\leadsto \mathsf{fma}\left(m, \color{blue}{\frac{m}{v}}, 0\right) \]

   10. Simplified 88.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(m, \frac{m}{v}, 0\right)} \]
   11. Step-by-step derivation

       1. +-rgt-identity N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. /-lowering-/.f64 88.0

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

   12. Applied egg-rr 88.0%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -1 \cdot 10^{-308}:\\ \;\;\;\;0 - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 8: 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 99.9%

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. metadata-eval 99.8

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

4. Applied egg-rr 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 9: 27.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 0 - m \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (- 0.0 m))

C

double code(double m, double v) {
	return 0.0 - m;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = 0.0d0 - m
end function

Java

public static double code(double m, double v) {
	return 0.0 - m;
}

Python

def code(m, v):
	return 0.0 - m

Julia

function code(m, v)
	return Float64(0.0 - m)
end

MATLAB

function tmp = code(m, v)
	tmp = 0.0 - m;
end

Wolfram

code[m_, v_] := N[(0.0 - m), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - m} \]

   3. --lowering--.f64 30.4

      \[\leadsto \color{blue}{0 - m} \]

5. Simplified 30.4%

   \[\leadsto \color{blue}{0 - m} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 30.4

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

7. Applied egg-rr 30.4%

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

8. Final simplification 30.4%

   \[\leadsto 0 - m \]
9. Add Preprocessing

Alternative 10: 3.0% accurate, 28.0× 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 m
end

MATLAB

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

Wolfram

code[m_, v_] := m

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - m} \]

   3. --lowering--.f64 30.4

      \[\leadsto \color{blue}{0 - m} \]

5. Simplified 30.4%

   \[\leadsto \color{blue}{0 - m} \]
6. Step-by-step derivation

   1. flip3-- N/A

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

   2. metadata-eval N/A

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

   3. sub0-neg N/A

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

   4. cube-neg N/A

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

   5. sub0-neg N/A

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

   6. sqr-pow N/A

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

   7. unpow-prod-down N/A

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

   8. sub0-neg N/A

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

   9. sub0-neg N/A

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

   10. sqr-neg N/A

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

   11. unpow-prod-down N/A

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

   12. sqr-pow N/A

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

   13. metadata-eval N/A

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

   14. +-lft-identity N/A

       \[\leadsto \frac{{m}^{3}}{\color{blue}{m \cdot m + 0 \cdot m}} \]

   15. mul0-lft N/A

       \[\leadsto \frac{{m}^{3}}{m \cdot m + \color{blue}{0}} \]

   16. +-rgt-identity N/A

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

   17. pow2 N/A

       \[\leadsto \frac{{m}^{3}}{\color{blue}{{m}^{2}}} \]

   18. pow-div N/A

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

   19. metadata-eval N/A

       \[\leadsto {m}^{\color{blue}{1}} \]

   20. unpow1 3.3

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

7. Applied egg-rr 3.3%

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

Reproduce

herbie shell --seed 2024199 
(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))