b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.3s

Alternatives: 11

Speedup: 1.0×

• 40.9% 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: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{-16}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5.2e-16)
   (* (- 1.0 m) (+ (/ m v) -1.0))
   (* (- 1.0 m) (* (- 1.0 m) (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.2e-16) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (1.0 - m) * ((1.0 - 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 <= 5.2d-16) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = (1.0d0 - m) * ((1.0d0 - m) * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5.2e-16) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (1.0 - m) * ((1.0 - m) * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.2e-16:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = (1.0 - m) * ((1.0 - m) * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.2e-16)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(1.0 - m) * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.2e-16)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = (1.0 - m) * ((1.0 - m) * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.2e-16], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.1999999999999997e-16

   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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{m}{v}\right)}, 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, v\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   5. Simplified 100.0%

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

   if 5.1999999999999997e-16 < 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 inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)}, \mathsf{\_.f64}\left(1, m\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{1}{m \cdot v} + {m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{\frac{1}{m}}{v} + {m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{m}^{2} \cdot \frac{1}{m}}{v} + {m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(m \cdot m\right) \cdot \frac{1}{m}}{v} + {m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(m \cdot \frac{1}{m}\right)}{v} + {m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      7. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} + {m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} + {m}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} + \left(m \cdot m\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} + m \cdot \left(m \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} + m \cdot \left(m \cdot \frac{\mathsf{neg}\left(1\right)}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} + m \cdot \left(m \cdot \frac{-1}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} + m \cdot \frac{m \cdot -1}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} + m \cdot \frac{-1 \cdot m}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} + m \cdot \left(-1 \cdot \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, m\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{m}{v}\right)\right)\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      18. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} - \frac{m}{v}\right)\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      19. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   5. Simplified 99.9%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/r/ N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(1 - m\right), \color{blue}{\left(\frac{m}{\frac{v}{1 - m}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(\frac{\color{blue}{m}}{\frac{v}{1 - m}}\right)\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(\frac{m}{v} \cdot \color{blue}{\left(1 - m\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(\left(1 - m\right) \cdot \color{blue}{\frac{m}{v}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\left(1 - m\right), \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(\frac{\color{blue}{m}}{v}\right)\right)\right) \]

      10. /-lowering-/.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

   7. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{-16}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

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 * ((m + -2.0) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = m * (m * ((m + (-2.0d0)) / v))
    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 * ((m + -2.0) / v));
	}
	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 * ((m + -2.0) / v))
	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(m * Float64(Float64(m + -2.0) / v)));
	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 * ((m + -2.0) / v));
	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[(m * N[(N[(m + -2.0), $MachinePrecision] / v), $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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{m}{v}\right)}, 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 98.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, v\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   5. Simplified 98.2%

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

   if 1.6000000000000001 < 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. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1}{\frac{v}{1 - m}}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{m}{\frac{v}{1 - m}}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \left(\frac{v}{1 - m}\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \left(1 - m\right)\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      6. --lowering--.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(1, m\right)\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(m \cdot {\left(1 - m\right)}^{2}\right), \color{blue}{v}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left({\left(1 - m\right)}^{2}\right)\right), v\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(\left(1 - m\right) \cdot \left(1 - m\right)\right)\right), v\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{*.f64}\left(\left(1 - m\right), \left(1 - m\right)\right)\right), v\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(1 - m\right)\right)\right), v\right) \]

      6. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{\_.f64}\left(1, m\right)\right)\right), v\right) \]

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf

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

   10. Simplified 98.0%

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

4. Final simplification 98.1%

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

Alternative 3: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m + -2}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (+ (/ m v) -1.0) (* m (* m (/ (+ m -2.0) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * (m * ((m + -2.0) / 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 <= 2.4d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = m * (m * ((m + (-2.0d0)) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * (m * ((m + -2.0) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = (m / v) + -1.0
	else:
		tmp = m * (m * ((m + -2.0) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(m * Float64(m * Float64(Float64(m + -2.0) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = (m / v) + -1.0;
	else
		tmp = m * (m * ((m + -2.0) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(m * N[(m * N[(N[(m + -2.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(m \cdot \left(1 + \frac{1}{v}\right)\right)}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m \cdot 1 + \color{blue}{m \cdot \frac{1}{v}}\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \color{blue}{m} \cdot \frac{1}{v}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m \cdot 1}{\color{blue}{v}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m}{v}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

      10. /-lowering-/.f64 98.1%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

   5. Simplified 98.1%

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

   6. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{m}{v}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 98.1%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right) \]

   8. Simplified 98.1%

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

   if 2.39999999999999991 < 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. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1}{\frac{v}{1 - m}}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{m}{\frac{v}{1 - m}}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \left(\frac{v}{1 - m}\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \left(1 - m\right)\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      6. --lowering--.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(1, m\right)\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(m \cdot {\left(1 - m\right)}^{2}\right), \color{blue}{v}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left({\left(1 - m\right)}^{2}\right)\right), v\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(\left(1 - m\right) \cdot \left(1 - m\right)\right)\right), v\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{*.f64}\left(\left(1 - m\right), \left(1 - m\right)\right)\right), v\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(1 - m\right)\right)\right), v\right) \]

      6. --lowering--.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{\_.f64}\left(1, m\right)\right)\right), v\right) \]

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf

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

   10. Simplified 98.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m + -2}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1}{\frac{v}{1 - m}}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{m}{\frac{v}{1 - m}}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \left(\frac{v}{1 - m}\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \left(1 - m\right)\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   6. --lowering--.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(1, m\right)\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

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(m * Float64(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[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1 - m}{v} \cdot m\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{1 - m}{v}\right), m\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 - m\right), v\right), m\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   5. --lowering--.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, m\right), v\right), m\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 7: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.38) (+ (/ m v) -1.0) (* m (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * (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 <= 0.38d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = m * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.38:
		tmp = (m / v) + -1.0
	else:
		tmp = m * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.38)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(m * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.38)
		tmp = (m / v) + -1.0;
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.38], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.38

   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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(m \cdot \left(1 + \frac{1}{v}\right)\right)}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m \cdot 1 + \color{blue}{m \cdot \frac{1}{v}}\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \color{blue}{m} \cdot \frac{1}{v}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m \cdot 1}{\color{blue}{v}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m}{v}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

      10. /-lowering-/.f64 98.1%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

   5. Simplified 98.1%

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

   6. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{m}{v}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 98.1%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right) \]

   8. Simplified 98.1%

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

   if 0.38 < 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. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1}{\frac{v}{1 - m}}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{m}{\frac{v}{1 - m}}\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \left(\frac{v}{1 - m}\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \left(1 - m\right)\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

      6. --lowering--.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(1, m\right)\right)\right), 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(m, \color{blue}{\left(\frac{{m}^{2}}{v}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{/.f64}\left(\left({m}^{2}\right), \color{blue}{v}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{/.f64}\left(\left(m \cdot m\right), v\right)\right) \]

      7. *-lowering-*.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, m\right), v\right)\right) \]

   7. Simplified 96.5%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(m, \left(\frac{m}{v} \cdot \color{blue}{m}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{*.f64}\left(\left(\frac{m}{v}\right), \color{blue}{m}\right)\right) \]

      3. /-lowering-/.f64 96.6%

         \[\leadsto \mathsf{*.f64}\left(m, \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, v\right), m\right)\right) \]

   9. Applied egg-rr 96.6%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.85 \cdot 10^{-171}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.85e-171) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.85e-171) {
		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 (m <= 2.85d-171) 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 (m <= 2.85e-171) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.85e-171:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.85e-171)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.85e-171)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.85e-171], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.8499999999999998e-171

   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 77.6%

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

   if 2.8499999999999998e-171 < 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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(m \cdot \left(1 + \frac{1}{v}\right)\right)}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m \cdot 1 + \color{blue}{m \cdot \frac{1}{v}}\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \color{blue}{m} \cdot \frac{1}{v}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m \cdot 1}{\color{blue}{v}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m}{v}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

      10. /-lowering-/.f64 69.1%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

   5. Simplified 69.1%

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

   6. Taylor expanded in v around 0

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

      1. /-lowering-/.f64 63.5%

         \[\leadsto \mathsf{/.f64}\left(m, \color{blue}{v}\right) \]

   8. Simplified 63.5%

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

Alternative 9: 75.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(m \cdot \left(1 + \frac{1}{v}\right)\right)}\right) \]

   5. distribute-lft-in N/A

      \[\leadsto \mathsf{+.f64}\left(-1, \left(m \cdot 1 + \color{blue}{m \cdot \frac{1}{v}}\right)\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \color{blue}{m} \cdot \frac{1}{v}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m \cdot 1}{\color{blue}{v}}\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{+.f64}\left(-1, \left(m + \frac{m}{v}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \color{blue}{\left(\frac{m}{v}\right)}\right)\right) \]

   10. /-lowering-/.f64 77.7%

       \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{+.f64}\left(m, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right)\right) \]

5. Simplified 77.7%

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

6. Taylor expanded in v around 0

   \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{m}{v}\right)}\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 77.7%

      \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(m, \color{blue}{v}\right)\right) \]

8. Simplified 77.7%

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

9. Final simplification 77.7%

   \[\leadsto \frac{m}{v} + -1 \]
10. Add Preprocessing

Alternative 10: 27.0% accurate, 4.3× 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

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

   2. neg-sub0 N/A

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

   3. associate--r- N/A

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

   4. metadata-eval N/A

      \[\leadsto -1 + m \]

   5. +-lowering-+.f64 27.2%

      \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{m}\right) \]

5. Simplified 27.2%

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

6. Final simplification 27.2%

   \[\leadsto m + -1 \]
7. Add Preprocessing

Alternative 11: 24.6% accurate, 13.0× 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 24.9%

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

Reproduce

herbie shell --seed 2024161 
(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)))