b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.8%

Time: 8.7s

Alternatives: 12

Speedup: 1.0×

• 41.3% 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

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

FPCore

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

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (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) * (1.0d0 - m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (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) * (1.0d0 - m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.1 \cdot 10^{-15}:\\ \;\;\;\;m + \mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m \cdot \left(m + -2\right), m, m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.1e-15)
   (+ m (fma (/ m v) (fma m -2.0 1.0) -1.0))
   (/ (fma (* m (+ m -2.0)) m m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.1e-15) {
		tmp = m + fma((m / v), fma(m, -2.0, 1.0), -1.0);
	} else {
		tmp = fma((m * (m + -2.0)), m, m) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.1e-15)
		tmp = Float64(m + fma(Float64(m / v), fma(m, -2.0, 1.0), -1.0));
	else
		tmp = Float64(fma(Float64(m * Float64(m + -2.0)), m, m) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.1e-15], N[(m + N[(N[(m / v), $MachinePrecision] * N[(m * -2.0 + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision] * m + m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.0999999999999999e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   if 3.0999999999999999e-15 < m

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in m around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Simplified 99.9%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 99.9

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

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(m \cdot m, m + -2, m\right)}{v}} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 99.9

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

   12. Applied egg-rr 99.9%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq 5 \cdot 10^{+15}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m \cdot m, m + -2, m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- 1.0 m) (+ (/ (* m (- 1.0 m)) v) -1.0)) 5e+15)
   (+ -1.0 (+ m (/ m v)))
   (/ (fma (* m m) (+ m -2.0) m) v)))

C

double code(double m, double v) {
	double tmp;
	if (((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= 5e+15) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = fma((m * m), (m + -2.0), m) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(1.0 - m) * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= 5e+15)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(fma(Float64(m * m), Float64(m + -2.0), m) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(1.0 - m), $MachinePrecision] * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 5e+15], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m * m), $MachinePrecision] * N[(m + -2.0), $MachinePrecision] + m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m)) < 5e15

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 5e15 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m))

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Simplified 99.8%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 99.9

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

   10. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq 5 \cdot 10^{+15}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m \cdot m, m + -2, m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- 1.0 m) (+ (/ (* m (- 1.0 m)) v) -1.0)) -0.5) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = 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 (((1.0d0 - m) * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if ((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= -0.5:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(1.0 - m) * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -0.5)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= -0.5)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(1.0 - m), $MachinePrecision] * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m)) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

   5. Simplified 93.7%

      \[\leadsto \color{blue}{-1} \]

   if -0.5 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m))

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 34.7

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

   5. Simplified 34.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 32.6

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

   8. Simplified 32.6%

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

   9. Taylor expanded in m around 0

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

       1. lower-/.f64 64.0

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

   11. Simplified 64.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.55 \cdot 10^{-26}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m \cdot \left(m + -2\right), m, m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.55e-26) (+ -1.0 (+ m (/ m v))) (/ (fma (* m (+ m -2.0)) m m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.55e-26) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = fma((m * (m + -2.0)), m, m) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.55e-26)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(fma(Float64(m * Float64(m + -2.0)), m, m) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.55e-26], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision] * m + m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.54999999999999992e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.54999999999999992e-26 < m

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in m around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   7. Simplified 99.8%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f64 99.8

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

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(m \cdot m, m + -2, m\right)}{v}} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 99.9

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

   12. Applied egg-rr 99.9%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(m + -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- 1.0 m) (+ (/ m v) -1.0)) (* m (* (/ m v) (+ m -2.0)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m * ((m / v) * (m + -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = m * ((m / v) * (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m * ((m / v) * (m + -2.0));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = m * ((m / v) * (m + -2.0))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m * Float64(Float64(m / v) * Float64(m + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = m * ((m / v) * (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m / v), $MachinePrecision] * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 98.8

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

   5. Simplified 98.8%

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

   if 1.6000000000000001 < m

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]
   6. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. associate-/r* N/A

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

      14. associate-*l/ N/A

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

      15. lft-mult-inverse N/A

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

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

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

      17. associate-*r* N/A

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

   7. Simplified 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(m + -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return (1.0 - 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 = (1.0d0 - m) * (((m * (1.0d0 - m)) / v) + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 7: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 98.8

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

   5. Simplified 98.8%

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

   if 1 < m

   1. Initial program 99.8%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.4

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

   5. Simplified 98.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 98.6%

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

Alternative 8: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6) (+ -1.0 (+ m (/ m v))) (* (/ m v) (* m m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m / v) * (m * 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 <= 2.6d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m / v) * (m * m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m / v) * (m * m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m / v) * Float64(m * m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m / v) * (m * m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000009

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 98.7

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

   5. Simplified 98.7%

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

   if 2.60000000000000009 < m

   1. Initial program 99.8%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.4

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

   5. Simplified 98.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{+148}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, m, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6e+148) (+ -1.0 (+ m (/ m v))) (fma m m -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6e+148) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = fma(m, m, -1.0);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6e+148)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = fma(m, m, -1.0);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6e+148], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * m + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6e148

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 73.9

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

   5. Simplified 73.9%

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

   if 1.6e148 < m

   1. Initial program 100.0%

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

   3. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 7.0

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

   5. Simplified 7.0%

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

      1. flip-+ N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 96.9

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

   7. Applied egg-rr 96.9%

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

   8. Taylor expanded in m around 0

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

   10. Simplified 97.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{+148}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, m, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. lower-+.f64 N/A

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

   10. lower-/.f64 73.1

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

5. Simplified 73.1%

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

Alternative 11: 27.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ m + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in v around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. associate--r- N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 22.6

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

5. Simplified 22.6%

   \[\leadsto \color{blue}{m + -1} \]
6. Add Preprocessing

Alternative 12: 25.1% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

5. Simplified 20.1%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (m v)
  :name "b 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) (- 1.0 m)))