b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.3s

Alternatives: 14

Speedup: 1.0×

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

Specification

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

Math

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

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. fma-define N/A

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

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

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

   5. fmm-undef N/A

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

   6. *-lft-identity N/A

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

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

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

   8. *-lowering-*.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-12)
   (+ (/ (* m (- 1.0 m)) v) -1.0)
   (* (- 1.0 m) (/ (- 1.0 m) (/ v m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-12) {
		tmp = ((m * (1.0 - m)) / v) + -1.0;
	} else {
		tmp = (1.0 - m) * ((1.0 - m) / (v / m));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 5d-12) then
        tmp = ((m * (1.0d0 - m)) / v) + (-1.0d0)
    else
        tmp = (1.0d0 - m) * ((1.0d0 - m) / (v / m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5e-12) {
		tmp = ((m * (1.0 - m)) / v) + -1.0;
	} else {
		tmp = (1.0 - m) * ((1.0 - m) / (v / m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5e-12:
		tmp = ((m * (1.0 - m)) / v) + -1.0
	else:
		tmp = (1.0 - m) * ((1.0 - m) / (v / m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-12)
		tmp = Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0);
	else
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(1.0 - m) / Float64(v / m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-12)
		tmp = ((m * (1.0 - m)) / v) + -1.0;
	else
		tmp = (1.0 - m) * ((1.0 - m) / (v / m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-12], N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.9999999999999997e-12

   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

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

   5. Simplified 99.6%

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

   if 4.9999999999999997e-12 < 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. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-/l* N/A

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

      4. div-sub N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. div-sub N/A

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

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

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

      16. --lowering--.f64 99.9%

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

   5. Simplified 99.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

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

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

      7. associate-/r/ N/A

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

      8. *-commutative N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

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

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

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

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

      13. /-lowering-/.f64 N/A

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

      14. --lowering--.f64 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{m \cdot \left(1 - m\right)}{v}\\ \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (/ (* m (- 1.0 m)) v)))
   (if (<= m 5e-12) (+ t_0 -1.0) (* (- 1.0 m) t_0))))

C

double code(double m, double v) {
	double t_0 = (m * (1.0 - m)) / v;
	double tmp;
	if (m <= 5e-12) {
		tmp = t_0 + -1.0;
	} else {
		tmp = (1.0 - m) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m * (1.0d0 - m)) / v
    if (m <= 5d-12) then
        tmp = t_0 + (-1.0d0)
    else
        tmp = (1.0d0 - m) * t_0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = (m * (1.0 - m)) / v;
	double tmp;
	if (m <= 5e-12) {
		tmp = t_0 + -1.0;
	} else {
		tmp = (1.0 - m) * t_0;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = (m * (1.0 - m)) / v
	tmp = 0
	if m <= 5e-12:
		tmp = t_0 + -1.0
	else:
		tmp = (1.0 - m) * t_0
	return tmp

Julia

function code(m, v)
	t_0 = Float64(Float64(m * Float64(1.0 - m)) / v)
	tmp = 0.0
	if (m <= 5e-12)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(Float64(1.0 - m) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = (m * (1.0 - m)) / v;
	tmp = 0.0;
	if (m <= 5e-12)
		tmp = t_0 + -1.0;
	else
		tmp = (1.0 - m) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]}, If[LessEqual[m, 5e-12], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 4.9999999999999997e-12

   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

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

   5. Simplified 99.6%

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

   if 4.9999999999999997e-12 < 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. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lft-mult-inverse N/A

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

      16. associate-*l/ N/A

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

      17. distribute-rgt-in N/A

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

      18. mul-1-neg N/A

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

   5. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \left(1 - m\right)\\ \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{t\_0}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1 - m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (- 1.0 m))))
   (if (<= m 5e-12) (+ (/ t_0 v) -1.0) (* t_0 (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double t_0 = m * (1.0 - m);
	double tmp;
	if (m <= 5e-12) {
		tmp = (t_0 / v) + -1.0;
	} else {
		tmp = t_0 * ((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) :: t_0
    real(8) :: tmp
    t_0 = m * (1.0d0 - m)
    if (m <= 5d-12) then
        tmp = (t_0 / v) + (-1.0d0)
    else
        tmp = t_0 * ((1.0d0 - m) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = m * (1.0 - m);
	double tmp;
	if (m <= 5e-12) {
		tmp = (t_0 / v) + -1.0;
	} else {
		tmp = t_0 * ((1.0 - m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (1.0 - m)
	tmp = 0
	if m <= 5e-12:
		tmp = (t_0 / v) + -1.0
	else:
		tmp = t_0 * ((1.0 - m) / v)
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(1.0 - m))
	tmp = 0.0
	if (m <= 5e-12)
		tmp = Float64(Float64(t_0 / v) + -1.0);
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 - m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (1.0 - m);
	tmp = 0.0;
	if (m <= 5e-12)
		tmp = (t_0 / v) + -1.0;
	else
		tmp = t_0 * ((1.0 - m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 5e-12], N[(N[(t$95$0 / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 4.9999999999999997e-12

   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

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

   5. Simplified 99.6%

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

   if 4.9999999999999997e-12 < 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. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-/l* N/A

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

      4. div-sub N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. div-sub N/A

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

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

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

      16. --lowering--.f64 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1 - m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1 - m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-12)
   (* (- 1.0 m) (+ (/ m v) -1.0))
   (* (* m (- 1.0 m)) (/ (- 1.0 m) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-12) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m * (1.0 - 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 <= 5d-12) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = (m * (1.0d0 - m)) * ((1.0d0 - m) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5e-12) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m * (1.0 - m)) * ((1.0 - m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5e-12:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = (m * (1.0 - m)) * ((1.0 - m) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-12)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m * Float64(1.0 - m)) * Float64(Float64(1.0 - m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-12)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = (m * (1.0 - m)) * ((1.0 - m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-12], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.9999999999999997e-12

   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 99.6%

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

   5. Simplified 99.6%

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

   if 4.9999999999999997e-12 < 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. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-/l* N/A

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

      4. div-sub N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. div-sub N/A

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

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

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

      16. --lowering--.f64 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1 - m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 97.9%

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

   5. Simplified 97.9%

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

   if 1 < 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. *-commutative N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. clear-num N/A

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

      8. flip-- N/A

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

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

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

      10. sub-neg N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   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 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 98.2%

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

Alternative 7: 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 100.0%

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

3. Final simplification 100.0%

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

Alternative 8: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. --lowering--.f64 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 9: 97.8% accurate, 1.1× speedup

Math

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

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(Float64(m * 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[(N[(m * m), $MachinePrecision] / v), $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 97.9%

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

   5. Simplified 97.9%

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

   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 97.9%

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

   8. Simplified 97.9%

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

   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. *-commutative N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. clear-num N/A

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

      8. flip-- N/A

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

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

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

      10. sub-neg N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   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 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 98.2%

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

Alternative 10: 62.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.55 \cdot 10^{-166}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.55e-166) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.55e-166) {
		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 <= 2.55d-166) 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 <= 2.55e-166) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.55e-166:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.55e-166)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.55e-166)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.55e-166], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5500000000000001e-166

   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 81.2%

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

   if 2.5500000000000001e-166 < 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 68.3%

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

   5. Simplified 68.3%

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

   6. Taylor expanded in v around 0

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

      1. /-lowering-/.f64 60.6%

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

   8. Simplified 60.6%

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

Alternative 11: 26.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 6.5e-26) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.5e-26) {
		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 <= 6.5d-26) 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 <= 6.5e-26) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.5e-26:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.5e-26)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.5e-26)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.5e-26], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 6.5e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

   5. Simplified 54.9%

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

   if 6.5e-26 < 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 55.6%

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

   5. Simplified 55.6%

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

   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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-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 54.8%

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

   8. Simplified 54.8%

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

   9. Taylor expanded in v around inf

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

   11. Simplified 5.8%

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

Alternative 12: 75.7% 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 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 76.7%

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

5. Simplified 76.7%

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

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 76.7%

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

8. Simplified 76.7%

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

9. Final simplification 76.7%

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

Alternative 13: 26.7% 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 100.0%

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

3. Taylor expanded in v around inf

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

   1. mul-1-neg N/A

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

   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 29.0%

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

5. Simplified 29.0%

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

6. Final simplification 29.0%

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

Alternative 14: 24.2% 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 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 26.5%

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

Reproduce

herbie shell --seed 2024150 
(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)))