b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.8%

Time: 8.8s

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6e-15)
   (* (- 1.0 m) (+ (/ m v) -1.0))
   (/ (* (- 1.0 m) (* m (- 1.0 m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6e-15) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = ((1.0 - m) * (m * (1.0 - 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 <= 2.6d-15) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = ((1.0d0 - m) * (m * (1.0d0 - m))) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.6e-15) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = ((1.0 - m) * (m * (1.0 - m))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6e-15:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = ((1.0 - m) * (m * (1.0 - m))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6e-15)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(m * Float64(1.0 - m))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6e-15)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = ((1.0 - m) * (m * (1.0 - m))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6e-15], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000004e-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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{m}{v}\right)}, 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]
   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 2.60000000000000004e-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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\frac{m \cdot {\left(1 - m\right)}^{2}}{v}} \]
   4. 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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.7 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{\left(m + -1\right) \cdot \left(m + -1\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.7e-15)
   (* (- 1.0 m) (+ (/ m v) -1.0))
   (* m (/ (* (+ m -1.0) (+ m -1.0)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.7e-15) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m * (((m + -1.0) * (m + -1.0)) / 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 <= 2.7d-15) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = m * (((m + (-1.0d0)) * (m + (-1.0d0))) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.7e-15) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m * (((m + -1.0) * (m + -1.0)) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.7e-15:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = m * (((m + -1.0) * (m + -1.0)) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.7e-15)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m * Float64(Float64(Float64(m + -1.0) * Float64(m + -1.0)) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.7e-15)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = m * (((m + -1.0) * (m + -1.0)) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.7e-15], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(N[(m + -1.0), $MachinePrecision] * N[(m + -1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.70000000000000009e-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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{m}{v}\right)}, 1\right), \mathsf{\_.f64}\left(1, m\right)\right) \]
   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 2.70000000000000009e-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

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\frac{m \cdot {\left(1 - m\right)}^{2}}{v}} \]
   4. 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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in v around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-in N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. sqr-neg N/A

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

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

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

      19. sub-neg N/A

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

      20. metadata-eval N/A

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

      21. +-commutative N/A

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

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

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

      23. sub-neg N/A

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

      24. metadata-eval N/A

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

      25. +-commutative N/A

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

      26. +-lowering-+.f64 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.7 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{\left(m + -1\right) \cdot \left(m + -1\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.65:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(m + -2\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.65) (* (- 1.0 m) (+ (/ m v) -1.0)) (* m (/ (* m (+ m -2.0)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.65) {
		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 <= 1.65d0) 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 <= 1.65) {
		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 <= 1.65:
		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 <= 1.65)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m * Float64(Float64(m * Float64(m + -2.0)) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.65)
		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, 1.65], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6499999999999999

   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

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

      1. /-lowering-/.f64 96.2%

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

   5. Simplified 96.2%

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

   if 1.6499999999999999 < 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 v around 0

      \[\leadsto \color{blue}{\frac{m \cdot {\left(1 - m\right)}^{2}}{v}} \]
   4. 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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   5. Simplified 99.9%

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

      1. clear-num N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in v around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. --lowering--.f64 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. associate-*r/ N/A

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

       7. *-rgt-identity N/A

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

       8. associate-/r* N/A

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

       9. associate-*r/ N/A

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

       10. associate-*r/ N/A

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

       11. div-sub N/A

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

       12. *-rgt-identity N/A

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

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

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

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

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

   13. Simplified 98.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.65:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(m + -2\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.9× speedup

Math

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

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(Float64(m * 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[(N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   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 96.1%

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

   5. Simplified 96.1%

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

   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 96.1%

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

   8. Simplified 96.1%

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

   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. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\frac{m \cdot {\left(1 - m\right)}^{2}}{v}} \]
   4. 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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 99.9%

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

   5. Simplified 99.9%

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

      1. clear-num N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in v around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. --lowering--.f64 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. associate-*r/ N/A

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

       7. *-rgt-identity N/A

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

       8. associate-/r* N/A

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

       9. associate-*r/ N/A

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

       10. associate-*r/ N/A

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

       11. div-sub N/A

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

       12. *-rgt-identity N/A

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

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

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

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

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

   13. Simplified 98.8%

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

4. Final simplification 97.6%

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

Alternative 5: 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 6: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.39:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.39) (+ (/ m v) -1.0) (* (/ m v) (* m m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.39) {
		tmp = (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 <= 0.39d0) then
        tmp = (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 <= 0.39) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.39:
		tmp = (m / v) + -1.0
	else:
		tmp = (m / v) * (m * m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.39)
		tmp = 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 <= 0.39)
		tmp = (m / v) + -1.0;
	else
		tmp = (m / v) * (m * m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.39], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.39000000000000001

   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 96.1%

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

   5. Simplified 96.1%

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

   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 96.1%

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

   8. Simplified 96.1%

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

   if 0.39000000000000001 < 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

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. 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. unpow2 N/A

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

      6. associate-/l* N/A

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

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

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

      8. /-lowering-/.f64 97.3%

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

   5. Simplified 97.3%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 97.3%

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

   7. Applied egg-rr 97.3%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.39:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% 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 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 96.1%

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

   5. Simplified 96.1%

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

   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 96.1%

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

   8. Simplified 96.1%

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

   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. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. 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. unpow2 N/A

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

      6. associate-/l* N/A

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

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

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

      8. /-lowering-/.f64 97.3%

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

   5. Simplified 97.3%

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

4. Final simplification 96.7%

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

Alternative 8: 62.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-174}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.9e-174) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-174) {
		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 <= 2.9d-174) 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 <= 2.9e-174) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.9e-174:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.9e-174)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.9e-174)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.9e-174], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.9000000000000001e-174

   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 81.9%

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

   if 2.9000000000000001e-174 < 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 v around 0

      \[\leadsto \color{blue}{\frac{m \cdot {\left(1 - m\right)}^{2}}{v}} \]
   4. 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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. --lowering--.f64 90.1%

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

   5. Simplified 90.1%

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

   6. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 54.6%

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

   8. Simplified 54.6%

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

Alternative 9: 27.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-25}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.1e-25) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-25) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.1d-25) then
        tmp = -1.0d0
    else
        tmp = m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-25) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.1e-25:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.1e-25)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.1e-25)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.1e-25], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 2.10000000000000002e-25

   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 57.0%

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

   if 2.10000000000000002e-25 < 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 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 51.0%

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

   5. Simplified 51.0%

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

   6. Taylor expanded in m around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

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

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

      6. /-lowering-/.f64 51.0%

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

   8. Simplified 51.0%

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

   9. Taylor expanded in v around inf

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

   11. Simplified 5.3%

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

Alternative 10: 76.0% 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 71.5%

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

5. Simplified 71.5%

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

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 71.5%

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

8. Simplified 71.5%

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

9. Final simplification 71.5%

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

Alternative 11: 27.3% 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 26.7%

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

5. Simplified 26.7%

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

6. Final simplification 26.7%

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

Alternative 12: 24.9% 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 24.3%

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

Reproduce

herbie shell --seed 2024194 
(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)))