a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 9.8s

Alternatives: 10

Speedup: 1.0×

• 42.1% 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 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   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. Final simplification 99.9%

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

Alternative 2: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-166}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 3.9 \cdot 10^{-125}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-112}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{-v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6e-166)
   (- m)
   (if (<= m 3.9e-125)
     (* m (/ m v))
     (if (<= m 8e-112)
       (- m)
       (if (<= m 1.0) (/ m (/ v m)) (* m (/ m (- v))))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6e-166) {
		tmp = -m;
	} else if (m <= 3.9e-125) {
		tmp = m * (m / v);
	} else if (m <= 8e-112) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / 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 <= 6d-166) then
        tmp = -m
    else if (m <= 3.9d-125) then
        tmp = m * (m / v)
    else if (m <= 8d-112) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m / (v / 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 <= 6e-166) {
		tmp = -m;
	} else if (m <= 3.9e-125) {
		tmp = m * (m / v);
	} else if (m <= 8e-112) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = m * (m / -v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6e-166:
		tmp = -m
	elif m <= 3.9e-125:
		tmp = m * (m / v)
	elif m <= 8e-112:
		tmp = -m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = m * (m / -v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6e-166)
		tmp = Float64(-m);
	elseif (m <= 3.9e-125)
		tmp = Float64(m * Float64(m / v));
	elseif (m <= 8e-112)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(m * Float64(m / Float64(-v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6e-166)
		tmp = -m;
	elseif (m <= 3.9e-125)
		tmp = m * (m / v);
	elseif (m <= 8e-112)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = m * (m / -v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6e-166], (-m), If[LessEqual[m, 3.9e-125], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 8e-112], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / (-v)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if m < 6.0000000000000005e-166 or 3.89999999999999982e-125 < m < 7.9999999999999996e-112

   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 m around 0 80.2%

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

      1. neg-mul-1 80.2%

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

   7. Simplified 80.2%

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

   if 6.0000000000000005e-166 < m < 3.89999999999999982e-125

   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%

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

   4. Taylor expanded in m around inf 68.7%

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

   if 7.9999999999999996e-112 < m < 1

   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 m around 0 96.0%

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

   4. Taylor expanded in m around inf 86.6%

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

      1. associate-/r/ 86.7%

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

   6. Applied egg-rr 86.7%

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

   if 1 < 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* 99.9%

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

      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 m around 0 0.1%

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

      1. neg-mul-1 0.1%

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

      2. +-commutative 0.1%

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

      3. unsub-neg 0.1%

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

   7. Simplified 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. unpow2 0.1%

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

      3. sqrt-unprod 0.1%

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

      4. sqr-neg 0.1%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 78.0%

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

      7. neg-mul-1 78.0%

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

      8. times-frac 78.0%

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

   9. Applied egg-rr 78.0%

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

   10. Taylor expanded in m around 0 78.0%

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

       1. mul-1-neg 78.0%

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

       2. +-commutative 78.0%

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

       3. *-commutative 78.0%

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

       4. unpow2 78.0%

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

       5. associate-*r/ 78.0%

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

       6. associate-*r* 78.0%

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

       7. fma-define 78.0%

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

       8. *-commutative 78.0%

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

       9. neg-mul-1 78.0%

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

       10. distribute-frac-neg2 78.0%

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

       11. neg-mul-1 78.0%

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

       12. associate-/l/ 78.0%

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

       13. fma-define 78.0%

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

       14. *-commutative 78.0%

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

       15. mul-1-neg 78.0%

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

       16. distribute-rgt-out 78.0%

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

       17. +-commutative 78.0%

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

       18. associate-/l/ 78.0%

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

       19. neg-mul-1 78.0%

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

       20. distribute-frac-neg2 78.0%

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

       21. sub-neg 78.0%

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

   12. Simplified 78.0%

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

   13. Taylor expanded in m around inf 78.0%

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

       1. associate-*r/ 78.0%

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

       2. neg-mul-1 78.0%

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

   15. Simplified 78.0%

       \[\leadsto m \cdot \color{blue}{\frac{-m}{v}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-166}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 3.9 \cdot 10^{-125}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-112}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{-v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 1.0], N[(t$95$0 - m), $MachinePrecision], N[(m * N[(-1.0 - t$95$0), $MachinePrecision]), $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. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 86.6%

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

      1. neg-mul-1 86.6%

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

      2. +-commutative 86.6%

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

      3. unsub-neg 86.6%

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

   7. Simplified 86.6%

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

      1. add-sqr-sqrt 86.5%

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

      2. unpow2 86.5%

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

      3. sqrt-unprod 52.7%

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

      4. sqr-neg 52.7%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 47.3%

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

      7. neg-mul-1 47.3%

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

      8. times-frac 47.2%

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

   9. Applied egg-rr 47.2%

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

       1. *-commutative 47.2%

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

       2. clear-num 47.2%

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

       3. frac-times 47.2%

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

       4. *-un-lft-identity 47.2%

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

   11. Applied egg-rr 47.2%

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

       1. frac-2neg 47.2%

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

       2. *-commutative 47.2%

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

       3. distribute-lft-neg-in 47.2%

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

       4. metadata-eval 47.2%

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

       5. *-un-lft-identity 47.2%

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

       6. distribute-frac-neg 47.2%

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

       7. frac-2neg 47.2%

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

       8. div-inv 47.2%

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

       9. metadata-eval 47.2%

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

       10. frac-2neg 47.2%

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

       11. associate-/l/ 47.3%

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

       12. add-sqr-sqrt 0.0%

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

       13. sqrt-unprod 52.7%

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

       14. sqr-neg 52.7%

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

       15. sqrt-unprod 86.4%

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

       16. add-sqr-sqrt 86.6%

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

       17. distribute-frac-neg2 86.6%

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

       18. associate-/l/ 98.6%

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

       19. frac-2neg 98.6%

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

       20. metadata-eval 98.6%

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

       21. div-inv 98.6%

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

       22. frac-2neg 98.6%

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

   13. Applied egg-rr 98.6%

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

   if 1 < 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* 99.9%

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

      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 m around inf 97.8%

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

      1. neg-mul-1 97.8%

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

      2. distribute-neg-frac2 97.8%

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

   7. Simplified 97.8%

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

4. Final simplification 98.2%

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

Alternative 4: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(m * Float64(-1.0 - 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 * (m / v)) - m;
	else
		tmp = m * (-1.0 - (m / (v / m)));
	end
	tmp_2 = tmp;
end

Wolfram

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 86.6%

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

      1. neg-mul-1 86.6%

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

      2. +-commutative 86.6%

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

      3. unsub-neg 86.6%

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

   7. Simplified 86.6%

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

      1. add-sqr-sqrt 86.5%

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

      2. unpow2 86.5%

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

      3. sqrt-unprod 52.7%

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

      4. sqr-neg 52.7%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 47.3%

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

      7. neg-mul-1 47.3%

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

      8. times-frac 47.2%

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

   9. Applied egg-rr 47.2%

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

       1. *-commutative 47.2%

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

       2. clear-num 47.2%

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

       3. frac-times 47.2%

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

       4. *-un-lft-identity 47.2%

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

   11. Applied egg-rr 47.2%

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

       1. frac-2neg 47.2%

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

       2. *-commutative 47.2%

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

       3. distribute-lft-neg-in 47.2%

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

       4. metadata-eval 47.2%

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

       5. *-un-lft-identity 47.2%

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

       6. distribute-frac-neg 47.2%

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

       7. frac-2neg 47.2%

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

       8. div-inv 47.2%

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

       9. metadata-eval 47.2%

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

       10. frac-2neg 47.2%

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

       11. associate-/l/ 47.3%

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

       12. add-sqr-sqrt 0.0%

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

       13. sqrt-unprod 52.7%

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

       14. sqr-neg 52.7%

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

       15. sqrt-unprod 86.4%

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

       16. add-sqr-sqrt 86.6%

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

       17. distribute-frac-neg2 86.6%

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

       18. associate-/l/ 98.6%

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

       19. frac-2neg 98.6%

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

       20. metadata-eval 98.6%

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

       21. div-inv 98.6%

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

       22. frac-2neg 98.6%

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

   13. Applied egg-rr 98.6%

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

   if 1 < 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* 99.9%

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

      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 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in m around inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. distribute-neg-frac2 97.8%

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

   9. Simplified 97.8%

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

4. Final simplification 98.2%

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

Alternative 5: 87.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m * (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(m / Float64(-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 * (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[(m / (-v)), $MachinePrecision]), $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 98.6%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. 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. Taylor expanded in m around 0 0.1%

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

      1. neg-mul-1 0.1%

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

      2. +-commutative 0.1%

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

      3. unsub-neg 0.1%

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

   7. Simplified 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. unpow2 0.1%

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

      3. sqrt-unprod 0.1%

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

      4. sqr-neg 0.1%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 78.0%

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

      7. neg-mul-1 78.0%

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

      8. times-frac 78.0%

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

   9. Applied egg-rr 78.0%

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

   10. Taylor expanded in m around 0 78.0%

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

       1. mul-1-neg 78.0%

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

       2. +-commutative 78.0%

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

       3. *-commutative 78.0%

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

       4. unpow2 78.0%

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

       5. associate-*r/ 78.0%

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

       6. associate-*r* 78.0%

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

       7. fma-define 78.0%

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

       8. *-commutative 78.0%

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

       9. neg-mul-1 78.0%

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

       10. distribute-frac-neg2 78.0%

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

       11. neg-mul-1 78.0%

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

       12. associate-/l/ 78.0%

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

       13. fma-define 78.0%

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

       14. *-commutative 78.0%

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

       15. mul-1-neg 78.0%

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

       16. distribute-rgt-out 78.0%

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

       17. +-commutative 78.0%

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

       18. associate-/l/ 78.0%

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

       19. neg-mul-1 78.0%

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

       20. distribute-frac-neg2 78.0%

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

       21. sub-neg 78.0%

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

   12. Simplified 78.0%

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

   13. Taylor expanded in m around inf 78.0%

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

       1. associate-*r/ 78.0%

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

       2. neg-mul-1 78.0%

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

   15. Simplified 78.0%

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

4. Final simplification 88.0%

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

Alternative 6: 87.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 86.6%

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

      1. neg-mul-1 86.6%

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

      2. +-commutative 86.6%

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

      3. unsub-neg 86.6%

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

   7. Simplified 86.6%

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

      1. add-sqr-sqrt 86.5%

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

      2. unpow2 86.5%

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

      3. sqrt-unprod 52.7%

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

      4. sqr-neg 52.7%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 47.3%

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

      7. neg-mul-1 47.3%

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

      8. times-frac 47.2%

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

   9. Applied egg-rr 47.2%

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

       1. *-commutative 47.2%

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

       2. clear-num 47.2%

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

       3. frac-times 47.2%

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

       4. *-un-lft-identity 47.2%

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

   11. Applied egg-rr 47.2%

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

       1. frac-2neg 47.2%

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

       2. *-commutative 47.2%

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

       3. distribute-lft-neg-in 47.2%

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

       4. metadata-eval 47.2%

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

       5. *-un-lft-identity 47.2%

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

       6. distribute-frac-neg 47.2%

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

       7. frac-2neg 47.2%

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

       8. div-inv 47.2%

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

       9. metadata-eval 47.2%

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

       10. frac-2neg 47.2%

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

       11. associate-/l/ 47.3%

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

       12. add-sqr-sqrt 0.0%

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

       13. sqrt-unprod 52.7%

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

       14. sqr-neg 52.7%

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

       15. sqrt-unprod 86.4%

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

       16. add-sqr-sqrt 86.6%

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

       17. distribute-frac-neg2 86.6%

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

       18. associate-/l/ 98.6%

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

       19. frac-2neg 98.6%

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

       20. metadata-eval 98.6%

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

       21. div-inv 98.6%

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

       22. frac-2neg 98.6%

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

   13. Applied egg-rr 98.6%

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

   if 1 < 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* 99.9%

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

      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 m around 0 0.1%

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

      1. neg-mul-1 0.1%

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

      2. +-commutative 0.1%

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

      3. unsub-neg 0.1%

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

   7. Simplified 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. unpow2 0.1%

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

      3. sqrt-unprod 0.1%

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

      4. sqr-neg 0.1%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 78.0%

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

      7. neg-mul-1 78.0%

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

      8. times-frac 78.0%

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

   9. Applied egg-rr 78.0%

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

   10. Taylor expanded in m around 0 78.0%

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

       1. mul-1-neg 78.0%

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

       2. +-commutative 78.0%

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

       3. *-commutative 78.0%

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

       4. unpow2 78.0%

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

       5. associate-*r/ 78.0%

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

       6. associate-*r* 78.0%

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

       7. fma-define 78.0%

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

       8. *-commutative 78.0%

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

       9. neg-mul-1 78.0%

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

       10. distribute-frac-neg2 78.0%

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

       11. neg-mul-1 78.0%

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

       12. associate-/l/ 78.0%

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

       13. fma-define 78.0%

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

       14. *-commutative 78.0%

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

       15. mul-1-neg 78.0%

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

       16. distribute-rgt-out 78.0%

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

       17. +-commutative 78.0%

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

       18. associate-/l/ 78.0%

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

       19. neg-mul-1 78.0%

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

       20. distribute-frac-neg2 78.0%

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

       21. sub-neg 78.0%

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

   12. Simplified 78.0%

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

   13. Taylor expanded in m around inf 78.0%

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

       1. associate-*r/ 78.0%

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

       2. neg-mul-1 78.0%

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

   15. Simplified 78.0%

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

4. Final simplification 88.0%

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

Alternative 7: 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 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   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 8: 37.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.08 \cdot 10^{-150}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 1.08e-150) (* m (/ m v)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 1.08e-150) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 1.08d-150) then
        tmp = m * (m / v)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 1.08e-150) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 1.08e-150:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 1.08e-150)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 1.08e-150)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 1.08e-150], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 1.08000000000000003e-150

   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 48.1%

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

   4. Taylor expanded in m around inf 36.7%

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

   if 1.08000000000000003e-150 < v

   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. Taylor expanded in m around 0 39.1%

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

      1. neg-mul-1 39.1%

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

   7. Simplified 39.1%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.08 \cdot 10^{-150}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 8 \cdot 10^{-151}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 8e-151) (/ m (/ v m)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 8e-151) {
		tmp = m / (v / m);
	} else {
		tmp = -m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 8d-151) then
        tmp = m / (v / m)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 8e-151) {
		tmp = m / (v / m);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 8e-151:
		tmp = m / (v / m)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 8e-151)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 8e-151)
		tmp = m / (v / m);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 8e-151], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 7.9999999999999995e-151

   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 48.1%

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

   4. Taylor expanded in m around inf 36.7%

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

      1. associate-/r/ 36.7%

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

   6. Applied egg-rr 36.7%

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

   if 7.9999999999999995e-151 < v

   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. Taylor expanded in m around 0 39.1%

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

      1. neg-mul-1 39.1%

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

   7. Simplified 39.1%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 8 \cdot 10^{-151}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.6% 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 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   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 m around 0 26.0%

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

   1. neg-mul-1 26.0%

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

7. Simplified 26.0%

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

8. Final simplification 26.0%

   \[\leadsto -m \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(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))