a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.1%

Time: 6.4s

Alternatives: 11

Speedup: 1.0×

• 41.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-41}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{1 - m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5.5e-41) (* (- (/ m v) 1.0) m) (* (* m m) (/ (- 1.0 m) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.5e-41) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = (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 <= 5.5d-41) then
        tmp = ((m / v) - 1.0d0) * m
    else
        tmp = (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 <= 5.5e-41) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = (m * m) * ((1.0 - m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.5e-41:
		tmp = ((m / v) - 1.0) * m
	else:
		tmp = (m * m) * ((1.0 - m) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.5e-41)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * m);
	else
		tmp = Float64(Float64(m * m) * Float64(Float64(1.0 - m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.5e-41)
		tmp = ((m / v) - 1.0) * m;
	else
		tmp = (m * m) * ((1.0 - m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.5e-41], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], N[(N[(m * m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.50000000000000022e-41

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 5.50000000000000022e-41 < m

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. div-inv N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-41}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{1 - m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+306}:\\ \;\;\;\;\frac{\left(-m\right) \cdot m}{m}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* (- (/ (* (- 1.0 m) m) v) 1.0) m)))
   (if (<= t_0 -1e+306)
     (/ (* (- m) m) m)
     (if (<= t_0 -2e-308) (- m) (* (/ m v) m)))))

C

double code(double m, double v) {
	double t_0 = ((((1.0 - m) * m) / v) - 1.0) * m;
	double tmp;
	if (t_0 <= -1e+306) {
		tmp = (-m * m) / m;
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	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 = ((((1.0d0 - m) * m) / v) - 1.0d0) * m
    if (t_0 <= (-1d+306)) then
        tmp = (-m * m) / m
    else if (t_0 <= (-2d-308)) then
        tmp = -m
    else
        tmp = (m / v) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = ((((1.0 - m) * m) / v) - 1.0) * m;
	double tmp;
	if (t_0 <= -1e+306) {
		tmp = (-m * m) / m;
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = ((((1.0 - m) * m) / v) - 1.0) * m
	tmp = 0
	if t_0 <= -1e+306:
		tmp = (-m * m) / m
	elif t_0 <= -2e-308:
		tmp = -m
	else:
		tmp = (m / v) * m
	return tmp

Julia

function code(m, v)
	t_0 = Float64(Float64(Float64(Float64(Float64(1.0 - m) * m) / v) - 1.0) * m)
	tmp = 0.0
	if (t_0 <= -1e+306)
		tmp = Float64(Float64(Float64(-m) * m) / m);
	elseif (t_0 <= -2e-308)
		tmp = Float64(-m);
	else
		tmp = Float64(Float64(m / v) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = ((((1.0 - m) * m) / v) - 1.0) * m;
	tmp = 0.0;
	if (t_0 <= -1e+306)
		tmp = (-m * m) / m;
	elseif (t_0 <= -2e-308)
		tmp = -m;
	else
		tmp = (m / v) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+306], N[(N[((-m) * m), $MachinePrecision] / m), $MachinePrecision], If[LessEqual[t$95$0, -2e-308], (-m), N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.00000000000000002e306

   1. Initial program 100.0%

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

   3. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.8

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 62.7%

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

   if -1.00000000000000002e306 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 99.9%

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

   3. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.6

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

   5. Applied rewrites 70.6%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.5%

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

   3. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 87.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -1 \cdot 10^{+306}:\\ \;\;\;\;\frac{\left(-m\right) \cdot m}{m}\\ \mathbf{elif}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\left(\left(-m\right) \cdot m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* (- 1.0 m) m) v) 1.0) m) -5e+41)
   (* (* (- m) m) (/ m v))
   (* (- (/ m v) 1.0) m)))

C

double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41) {
		tmp = (-m * m) * (m / v);
	} else {
		tmp = ((m / v) - 1.0) * m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((((((1.0d0 - m) * m) / v) - 1.0d0) * m) <= (-5d+41)) then
        tmp = (-m * m) * (m / v)
    else
        tmp = ((m / v) - 1.0d0) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41) {
		tmp = (-m * m) * (m / v);
	} else {
		tmp = ((m / v) - 1.0) * m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41:
		tmp = (-m * m) * (m / v)
	else:
		tmp = ((m / v) - 1.0) * m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(1.0 - m) * m) / v) - 1.0) * m) <= -5e+41)
		tmp = Float64(Float64(Float64(-m) * m) * Float64(m / v));
	else
		tmp = Float64(Float64(Float64(m / v) - 1.0) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41)
		tmp = (-m * m) * (m / v);
	else
		tmp = ((m / v) - 1.0) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -5e+41], N[(N[((-m) * m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -5.00000000000000022e41

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. cube-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

   if -5.00000000000000022e41 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\left(\left(-m\right) \cdot m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(-m\right) \cdot m}{m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* (- 1.0 m) m) v) 1.0) m) -5e+41)
   (/ (* (- m) m) m)
   (* (- (/ m v) 1.0) m)))

C

double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41) {
		tmp = (-m * m) / m;
	} else {
		tmp = ((m / v) - 1.0) * m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((((((1.0d0 - m) * m) / v) - 1.0d0) * m) <= (-5d+41)) then
        tmp = (-m * m) / m
    else
        tmp = ((m / v) - 1.0d0) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41) {
		tmp = (-m * m) / m;
	} else {
		tmp = ((m / v) - 1.0) * m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41:
		tmp = (-m * m) / m
	else:
		tmp = ((m / v) - 1.0) * m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(1.0 - m) * m) / v) - 1.0) * m) <= -5e+41)
		tmp = Float64(Float64(Float64(-m) * m) / m);
	else
		tmp = Float64(Float64(Float64(m / v) - 1.0) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41)
		tmp = (-m * m) / m;
	else
		tmp = ((m / v) - 1.0) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -5e+41], N[(N[((-m) * m), $MachinePrecision] / m), $MachinePrecision], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -5.00000000000000022e41

   1. Initial program 99.9%

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

   3. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.6

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

   5. Applied rewrites 5.6%

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

   7. Applied rewrites 52.1%

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

   if -5.00000000000000022e41 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(-m\right) \cdot m}{m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(-m\right) \cdot m}{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{m - v}{v} \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* (- 1.0 m) m) v) 1.0) m) -5e+41)
   (/ (* (- m) m) m)
   (* (/ (- m v) v) m)))

C

double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41) {
		tmp = (-m * m) / m;
	} else {
		tmp = ((m - v) / 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 ((((((1.0d0 - m) * m) / v) - 1.0d0) * m) <= (-5d+41)) then
        tmp = (-m * m) / m
    else
        tmp = ((m - v) / v) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41) {
		tmp = (-m * m) / m;
	} else {
		tmp = ((m - v) / v) * m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41:
		tmp = (-m * m) / m
	else:
		tmp = ((m - v) / v) * m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(1.0 - m) * m) / v) - 1.0) * m) <= -5e+41)
		tmp = Float64(Float64(Float64(-m) * m) / m);
	else
		tmp = Float64(Float64(Float64(m - v) / v) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -5e+41)
		tmp = (-m * m) / m;
	else
		tmp = ((m - v) / v) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -5e+41], N[(N[((-m) * m), $MachinePrecision] / m), $MachinePrecision], N[(N[(N[(m - v), $MachinePrecision] / v), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -5.00000000000000022e41

   1. Initial program 99.9%

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

   3. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.6

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

   5. Applied rewrites 5.6%

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

   7. Applied rewrites 52.1%

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

   if -5.00000000000000022e41 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.7%

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

   3. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in m around 0

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

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 97.2

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

   8. Applied rewrites 97.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(-m\right) \cdot m}{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{m - v}{v} \cdot m\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* (- 1.0 m) m) v) 1.0) m) -2e-308) (- m) (* (/ m v) m)))

C

double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -2e-308) {
		tmp = -m;
	} else {
		tmp = (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 ((((((1.0d0 - m) * m) / v) - 1.0d0) * m) <= (-2d-308)) then
        tmp = -m
    else
        tmp = (m / v) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -2e-308) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (((((1.0 - m) * m) / v) - 1.0) * m) <= -2e-308:
		tmp = -m
	else:
		tmp = (m / v) * m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(1.0 - m) * m) / v) - 1.0) * m) <= -2e-308)
		tmp = Float64(-m);
	else
		tmp = Float64(Float64(m / v) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((((((1.0 - m) * m) / v) - 1.0) * m) <= -2e-308)
		tmp = -m;
	else
		tmp = (m / v) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -2e-308], (-m), N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 99.9%

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

   3. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 34.2

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

   5. Applied rewrites 34.2%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.5%

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

   3. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 87.2%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right) \cdot m \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-41}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5.5e-41) (* (- (/ m v) 1.0) m) (/ (* (* (- 1.0 m) m) m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.5e-41) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = (((1.0 - 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 <= 5.5d-41) then
        tmp = ((m / v) - 1.0d0) * m
    else
        tmp = (((1.0d0 - m) * m) * m) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5.5e-41) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = (((1.0 - m) * m) * m) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.5e-41:
		tmp = ((m / v) - 1.0) * m
	else:
		tmp = (((1.0 - m) * m) * m) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.5e-41)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * m);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) * m) * m) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.5e-41)
		tmp = ((m / v) - 1.0) * m;
	else
		tmp = (((1.0 - m) * m) * m) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.5e-41], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.50000000000000022e-41

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 5.50000000000000022e-41 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 99.9

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

   10. Applied rewrites 99.9%

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

Alternative 8: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. cube-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

4. Final simplification 97.6%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 10: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 11: 27.2% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := (-m)

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 26.2

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

5. Applied rewrites 26.2%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) m))