a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 22.6s

Alternatives: 12

Speedup: N/A×

• 42.0% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.02 \cdot 10^{-159}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 - \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.02e-159)
   (- m)
   (if (<= m 1.0) (* m (/ m v)) (* m (- -1.0 (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.02e-159) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = 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 <= 1.02d-159) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m * (m / v)
    else
        tmp = m * ((-1.0d0) - (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.02e-159) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = m * (-1.0 - (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.02e-159:
		tmp = -m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = m * (-1.0 - (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.02e-159)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(m * Float64(-1.0 - Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.02e-159)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = m * (-1.0 - (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.02e-159], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(-1.0 - N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.02e-159

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 76.4%

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

      1. neg-mul-1 76.4%

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

   9. Simplified 76.4%

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

   if 1.02e-159 < m < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0 91.1%

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

   4. Taylor expanded in m around inf 74.4%

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

   if 1 < m

   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 m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. fma-neg 0.1%

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

      3. frac-2neg 0.1%

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

      4. metadata-eval 0.1%

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

      5. mul-1-neg 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 83.4%

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

      8. mul-1-neg 83.4%

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

      9. mul-1-neg 83.4%

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

      10. sqr-neg 83.4%

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

      11. sqrt-unprod 81.2%

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

      12. add-sqr-sqrt 81.2%

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

      13. metadata-eval 81.2%

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

   5. Applied egg-rr 81.2%

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

      1. fma-undefine 81.2%

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

      2. associate-*r/ 81.2%

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

      3. associate-*l/ 81.2%

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

      4. *-commutative 81.2%

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

      5. +-commutative 81.2%

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

      6. mul-1-neg 81.2%

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

      7. unsub-neg 81.2%

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

   7. Simplified 81.2%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.02 \cdot 10^{-159}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 - \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-159}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-159) (- m) (if (<= m 1.0) (* m (/ m v)) (* m (/ (- m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-159) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = 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 <= 5d-159) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m * (m / v)
    else
        tmp = m * (-m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5e-159) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = m * (-m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5e-159:
		tmp = -m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = m * (-m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-159)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(m * Float64(Float64(-m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-159)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = m * (-m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-159], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[((-m) / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 5.00000000000000032e-159

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 76.4%

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

      1. neg-mul-1 76.4%

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

   9. Simplified 76.4%

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

   if 5.00000000000000032e-159 < m < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0 91.1%

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

   4. Taylor expanded in m around inf 74.4%

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

   if 1 < m

   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 m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. fma-neg 0.1%

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

      3. frac-2neg 0.1%

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

      4. metadata-eval 0.1%

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

      5. mul-1-neg 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 83.4%

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

      8. mul-1-neg 83.4%

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

      9. mul-1-neg 83.4%

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

      10. sqr-neg 83.4%

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

      11. sqrt-unprod 81.2%

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

      12. add-sqr-sqrt 81.2%

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

      13. metadata-eval 81.2%

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

   5. Applied egg-rr 81.2%

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

      1. fma-undefine 81.2%

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

      2. associate-*r/ 81.2%

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

      3. associate-*l/ 81.2%

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

      4. *-commutative 81.2%

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

      5. +-commutative 81.2%

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

      6. mul-1-neg 81.2%

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

      7. unsub-neg 81.2%

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

   7. Simplified 81.2%

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

   8. Taylor expanded in m around inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. distribute-frac-neg2 81.2%

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

   10. Simplified 81.2%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-159}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.95 \cdot 10^{-159} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 1.95e-159) (not (<= m 1.0))) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 1.95e-159) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = 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 <= 1.95d-159) .or. (.not. (m <= 1.0d0))) then
        tmp = -m
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m <= 1.95e-159) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 1.95e-159) or not (m <= 1.0):
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((m <= 1.95e-159) || !(m <= 1.0))
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m <= 1.95e-159) || ~((m <= 1.0)))
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 1.95e-159], N[Not[LessEqual[m, 1.0]], $MachinePrecision]], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.94999999999999988e-159 or 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 33.6%

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

      1. neg-mul-1 33.6%

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

   9. Simplified 33.6%

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

   if 1.94999999999999988e-159 < m < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in m around 0 91.1%

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

   4. Taylor expanded in m around inf 74.4%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.95 \cdot 10^{-159} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.45 \cdot 10^{-65}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.45e-65) (* m (+ -1.0 (/ m v))) (/ m (/ v (* m (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.45e-65) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m / (v / (m * (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 (m <= 2.45d-65) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = m / (v / (m * (1.0d0 - m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.45e-65) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m / (v / (m * (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.45e-65:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m / (v / (m * (1.0 - m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.45e-65)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m / Float64(v / Float64(m * Float64(1.0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.45e-65)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = m / (v / (m * (1.0 - m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.45e-65], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.44999999999999982e-65

   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 0 99.9%

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

   if 2.44999999999999982e-65 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

   7. Simplified 99.9%

      \[\leadsto \color{blue}{{m}^{2} \cdot \frac{1 - m}{v}} \]
   8. Step-by-step derivation

      1. unpow2 99.9%

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

      2. associate-*l* 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r/ 99.8%

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

      5. clear-num 99.8%

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

      6. un-div-inv 99.9%

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

      7. associate-/l/ 99.9%

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

      8. *-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.45 \cdot 10^{-65}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.4 \cdot 10^{-15}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.4e-15) (* m (+ -1.0 (/ m v))) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.4e-15) {
		tmp = m * (-1.0 + (m / v));
	} 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 <= 6.4d-15) then
        tmp = m * ((-1.0d0) + (m / v))
    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 <= 6.4e-15) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.4e-15:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.4e-15)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.4e-15)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.4e-15], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.3999999999999999e-15

   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 99.8%

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

   if 6.3999999999999999e-15 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fma-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in v around 0 99.9%

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

      1. associate-*r/ 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.4 \cdot 10^{-15}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 95.9%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 99.9%

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

   7. Simplified 99.9%

      \[\leadsto \color{blue}{{m}^{2} \cdot \frac{1 - m}{v}} \]
   8. Step-by-step derivation

      1. unpow2 99.9%

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

      2. associate-*l* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r/ 99.9%

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

      5. associate-*r/ 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in m around inf 98.0%

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

       1. neg-mul-1 98.0%

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

   12. Simplified 98.0%

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

4. Final simplification 96.9%

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

Alternative 8: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 95.9%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 99.9%

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

   7. Simplified 99.9%

      \[\leadsto \color{blue}{{m}^{2} \cdot \frac{1 - m}{v}} \]
   8. Step-by-step derivation

      1. unpow2 99.9%

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

      2. associate-*l* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r/ 99.9%

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

      5. clear-num 99.9%

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

      6. un-div-inv 99.9%

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

      7. associate-/l/ 100.0%

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

      8. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in m around inf 98.0%

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

       1. neg-mul-1 98.0%

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

   12. Simplified 98.0%

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

4. Final simplification 96.8%

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

Alternative 9: 87.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 95.9%

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

   if 1 < m

   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 m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. fma-neg 0.1%

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

      3. frac-2neg 0.1%

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

      4. metadata-eval 0.1%

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

      5. mul-1-neg 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 83.4%

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

      8. mul-1-neg 83.4%

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

      9. mul-1-neg 83.4%

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

      10. sqr-neg 83.4%

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

      11. sqrt-unprod 81.2%

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

      12. add-sqr-sqrt 81.2%

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

      13. metadata-eval 81.2%

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

   5. Applied egg-rr 81.2%

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

      1. fma-undefine 81.2%

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

      2. associate-*r/ 81.2%

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

      3. associate-*l/ 81.2%

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

      4. *-commutative 81.2%

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

      5. +-commutative 81.2%

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

      6. mul-1-neg 81.2%

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

      7. unsub-neg 81.2%

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

   7. Simplified 81.2%

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

4. Final simplification 89.2%

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

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 11: 27.4% accurate, 5.5× 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 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in m around 0 28.9%

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

   1. neg-mul-1 28.9%

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

9. Simplified 28.9%

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

Alternative 12: 3.0% accurate, 11.0× 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 m
end

MATLAB

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

Wolfram

code[m_, v_] := m

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-/l* 99.9%

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

   4. fma-neg 99.9%

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

   5. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in m around 0 28.9%

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

   1. *-commutative 28.9%

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

   2. neg-mul-1 28.9%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 3.6%

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

   5. sqr-neg 3.6%

      \[\leadsto \sqrt{\color{blue}{m \cdot m}} \]

   6. sqrt-unprod 3.5%

      \[\leadsto \color{blue}{\sqrt{m} \cdot \sqrt{m}} \]

   7. add-sqr-sqrt 3.5%

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

   8. /-rgt-identity 3.5%

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

7. Applied egg-rr 3.5%

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

8. Taylor expanded in m around 0 3.5%

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

Reproduce

herbie shell --seed 2024141 
(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))