a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.6%

Time: 8.0s

Alternatives: 11

Speedup: 1.0×

• 42.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(1 - m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.25e-23) (- (/ m (/ v m)) m) (* (/ m v) (* m (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-23) {
		tmp = (m / (v / m)) - m;
	} 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 <= 1.25d-23) then
        tmp = (m / (v / m)) - m
    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 <= 1.25e-23) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (m / v) * (m * (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.25e-23:
		tmp = (m / (v / m)) - m
	else:
		tmp = (m / v) * (m * (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.25e-23)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(Float64(m / v) * Float64(m * Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.25e-23)
		tmp = (m / (v / m)) - m;
	else
		tmp = (m / v) * (m * (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.25e-23], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.2500000000000001e-23

   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* 99.7%

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

      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 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. neg-mul-1 99.8%

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

      6. unsub-neg 99.8%

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

      7. associate-/l* 85.3%

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

      8. unpow2 85.3%

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

   7. Simplified 85.3%

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

      1. unpow2 85.3%

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

      2. add-sqr-sqrt 85.1%

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

      3. sqrt-unprod 47.8%

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

      4. sqr-neg 47.8%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 55.5%

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

      7. associate-*l/ 55.4%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 48.3%

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

      10. sqr-neg 48.3%

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

      11. sqrt-unprod 99.5%

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

      12. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. *-commutative 99.8%

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

       2. clear-num 99.8%

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

       3. un-div-inv 99.8%

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

   11. Applied egg-rr 99.8%

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

   if 1.2500000000000001e-23 < 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 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 v around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.8%

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

      3. *-lft-identity 99.8%

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

      4. associate-*l/ 99.8%

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

      5. associate-/r/ 99.8%

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

      6. associate-*r/ 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-rgt1-in 99.9%

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

      10. *-rgt-identity 99.9%

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

      11. sub-neg 99.9%

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

   9. Simplified 99.9%

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

       1. clear-num 99.8%

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

       2. associate-/r/ 99.8%

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

       3. clear-num 99.8%

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

       4. unpow2 99.8%

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

       5. associate-*r/ 99.9%

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

       6. associate-*r* 99.9%

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

       7. /-rgt-identity 99.9%

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

       8. associate-/r/ 99.9%

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

       9. associate-/l/ 99.9%

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

       10. associate-*r/ 99.9%

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

       11. associate-*l/ 99.9%

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

       12. associate-/l/ 99.9%

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

       13. associate-/r/ 99.9%

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

       14. /-rgt-identity 99.9%

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

       15. associate-*l* 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(1 - m\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.85 \cdot 10^{-177}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{elif}\;m \leq 2.4 \cdot 10^{-125}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.85e-177)
   (- m)
   (if (<= m 2e-133)
     (/ m (/ v m))
     (if (<= m 2.4e-125)
       (- m)
       (if (<= m 1.0) (* m (/ m v)) (* (/ m v) (- m)))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.85e-177) {
		tmp = -m;
	} else if (m <= 2e-133) {
		tmp = m / (v / m);
	} else if (m <= 2.4e-125) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = (m / v) * -m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.85d-177) then
        tmp = -m
    else if (m <= 2d-133) then
        tmp = m / (v / m)
    else if (m <= 2.4d-125) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m * (m / v)
    else
        tmp = (m / v) * -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.85e-177) {
		tmp = -m;
	} else if (m <= 2e-133) {
		tmp = m / (v / m);
	} else if (m <= 2.4e-125) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = (m / v) * -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.85e-177:
		tmp = -m
	elif m <= 2e-133:
		tmp = m / (v / m)
	elif m <= 2.4e-125:
		tmp = -m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = (m / v) * -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.85e-177)
		tmp = Float64(-m);
	elseif (m <= 2e-133)
		tmp = Float64(m / Float64(v / m));
	elseif (m <= 2.4e-125)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(Float64(m / v) * Float64(-m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.85e-177)
		tmp = -m;
	elseif (m <= 2e-133)
		tmp = m / (v / m);
	elseif (m <= 2.4e-125)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = (m / v) * -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.85e-177], (-m), If[LessEqual[m, 2e-133], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.4e-125], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * (-m)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if m < 1.84999999999999997e-177 or 2.0000000000000001e-133 < m < 2.4000000000000001e-125

   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 81.8%

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

      1. neg-mul-1 81.8%

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

   7. Simplified 81.8%

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

   if 1.84999999999999997e-177 < m < 2.0000000000000001e-133

   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 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. neg-mul-1 99.7%

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

      6. unsub-neg 99.7%

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

      7. associate-/l* 66.1%

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

      8. unpow2 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in m around inf 34.6%

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

      1. *-un-lft-identity 34.6%

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

      2. add-sqr-sqrt 34.4%

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

      3. sqrt-unprod 2.3%

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

      4. sqr-neg 2.3%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 4.2%

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

      7. neg-mul-1 4.2%

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

      8. times-frac 4.2%

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

      9. metadata-eval 4.2%

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

      10. unpow2 4.2%

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

      11. add-sqr-sqrt 4.2%

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

      12. sqrt-unprod 2.4%

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

      13. sqr-neg 2.4%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 34.6%

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

      16. associate-*r/ 67.8%

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

      17. *-commutative 67.8%

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

      18. *-commutative 67.8%

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

      19. associate-*l* 67.8%

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

   10. Applied egg-rr 3.7%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 34.8%

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

       3. sqr-neg 34.8%

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

       4. sqrt-unprod 67.5%

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

       5. add-sqr-sqrt 67.8%

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

       6. associate-/r/ 67.9%

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

   12. Applied egg-rr 67.9%

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

   if 2.4000000000000001e-125 < 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* 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in m around 0 90.3%

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

      1. sub-neg 90.3%

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

      2. metadata-eval 90.3%

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

      3. distribute-lft-in 90.3%

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

      4. *-commutative 90.3%

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

      5. neg-mul-1 90.3%

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

      6. unsub-neg 90.3%

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

      7. associate-/l* 90.4%

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

      8. unpow2 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in m around inf 78.5%

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

      1. unpow2 90.4%

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

      2. add-sqr-sqrt 89.8%

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

      3. sqrt-unprod 44.2%

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

      4. sqr-neg 44.2%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 13.0%

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

      7. associate-*l/ 13.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 44.2%

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

      10. sqr-neg 44.2%

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

      11. sqrt-unprod 89.7%

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

      12. add-sqr-sqrt 90.3%

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

   10. Applied egg-rr 78.3%

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

   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%

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

      1. sub-neg 0.1%

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

      2. metadata-eval 0.1%

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

      3. distribute-lft-in 0.1%

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

      4. *-commutative 0.1%

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

      5. neg-mul-1 0.1%

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

      6. unsub-neg 0.1%

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

      7. associate-/l* 0.1%

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

      8. unpow2 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in m around inf 0.1%

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

      1. *-un-lft-identity 0.1%

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

      2. add-sqr-sqrt 0.1%

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

      3. sqrt-unprod 0.1%

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

      4. sqr-neg 0.1%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 82.5%

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

      7. neg-mul-1 82.5%

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

      8. times-frac 82.5%

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

      9. metadata-eval 82.5%

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

      10. unpow2 82.5%

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

      11. add-sqr-sqrt 82.5%

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

      12. sqrt-unprod 83.7%

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

      13. sqr-neg 83.7%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.1%

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

      16. associate-*r/ 0.1%

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

      17. *-commutative 0.1%

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

      18. *-commutative 0.1%

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

      19. associate-*l* 0.1%

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

   10. Applied egg-rr 82.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.85 \cdot 10^{-177}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{elif}\;m \leq 2.4 \cdot 10^{-125}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 35.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.25 \cdot 10^{-238} \lor \neg \left(v \leq 2.65 \cdot 10^{-204}\right) \land v \leq 4.9 \cdot 10^{-191}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= v 1.25e-238) (and (not (<= v 2.65e-204)) (<= v 4.9e-191)))
   (/ m (/ v m))
   (- m)))

C

double code(double m, double v) {
	double tmp;
	if ((v <= 1.25e-238) || (!(v <= 2.65e-204) && (v <= 4.9e-191))) {
		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 <= 1.25d-238) .or. (.not. (v <= 2.65d-204)) .and. (v <= 4.9d-191)) 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 <= 1.25e-238) || (!(v <= 2.65e-204) && (v <= 4.9e-191))) {
		tmp = m / (v / m);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (v <= 1.25e-238) or (not (v <= 2.65e-204) and (v <= 4.9e-191)):
		tmp = m / (v / m)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((v <= 1.25e-238) || (!(v <= 2.65e-204) && (v <= 4.9e-191)))
		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 <= 1.25e-238) || (~((v <= 2.65e-204)) && (v <= 4.9e-191)))
		tmp = m / (v / m);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[v, 1.25e-238], And[N[Not[LessEqual[v, 2.65e-204]], $MachinePrecision], LessEqual[v, 4.9e-191]]], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 1.25e-238 or 2.6499999999999999e-204 < v < 4.9e-191

   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* 99.7%

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

      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 51.4%

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

      1. sub-neg 51.4%

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

      2. metadata-eval 51.4%

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

      3. distribute-lft-in 51.4%

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

      4. *-commutative 51.4%

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

      5. neg-mul-1 51.4%

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

      6. unsub-neg 51.4%

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

      7. associate-/l* 32.7%

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

      8. unpow2 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in m around inf 29.4%

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

      1. *-un-lft-identity 29.4%

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

      2. add-sqr-sqrt 29.2%

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

      3. sqrt-unprod 1.2%

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

      4. sqr-neg 1.2%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 46.5%

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

      7. neg-mul-1 46.5%

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

      8. times-frac 46.5%

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

      9. metadata-eval 46.5%

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

      10. unpow2 46.5%

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

      11. add-sqr-sqrt 46.5%

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

      12. sqrt-unprod 45.2%

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

      13. sqr-neg 45.2%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 29.4%

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

      16. associate-*r/ 47.8%

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

      17. *-commutative 47.8%

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

      18. *-commutative 47.8%

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

      19. associate-*l* 47.8%

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

   10. Applied egg-rr 46.3%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 29.3%

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

       3. sqr-neg 29.3%

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

       4. sqrt-unprod 47.7%

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

       5. add-sqr-sqrt 47.8%

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

       6. associate-/r/ 47.9%

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

   12. Applied egg-rr 47.9%

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

   if 1.25e-238 < v < 2.6499999999999999e-204 or 4.9e-191 < 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 33.9%

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

      1. neg-mul-1 33.9%

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

   7. Simplified 33.9%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.25 \cdot 10^{-238} \lor \neg \left(v \leq 2.65 \cdot 10^{-204}\right) \land v \leq 4.9 \cdot 10^{-191}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 35.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.25 \cdot 10^{-238} \lor \neg \left(v \leq 9.2 \cdot 10^{-205}\right) \land v \leq 10^{-190}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= v 1.25e-238) (and (not (<= v 9.2e-205)) (<= v 1e-190)))
   (* m (/ m v))
   (- m)))

C

double code(double m, double v) {
	double tmp;
	if ((v <= 1.25e-238) || (!(v <= 9.2e-205) && (v <= 1e-190))) {
		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 <= 1.25d-238) .or. (.not. (v <= 9.2d-205)) .and. (v <= 1d-190)) 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 <= 1.25e-238) || (!(v <= 9.2e-205) && (v <= 1e-190))) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (v <= 1.25e-238) or (not (v <= 9.2e-205) and (v <= 1e-190)):
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((v <= 1.25e-238) || (!(v <= 9.2e-205) && (v <= 1e-190)))
		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 <= 1.25e-238) || (~((v <= 9.2e-205)) && (v <= 1e-190)))
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[v, 1.25e-238], And[N[Not[LessEqual[v, 9.2e-205]], $MachinePrecision], LessEqual[v, 1e-190]]], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 1.25e-238 or 9.1999999999999997e-205 < v < 1e-190

   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* 99.7%

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

      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 51.4%

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

      1. sub-neg 51.4%

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

      2. metadata-eval 51.4%

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

      3. distribute-lft-in 51.4%

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

      4. *-commutative 51.4%

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

      5. neg-mul-1 51.4%

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

      6. unsub-neg 51.4%

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

      7. associate-/l* 32.7%

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

      8. unpow2 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in m around inf 29.4%

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

      1. unpow2 32.7%

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

      2. add-sqr-sqrt 32.6%

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

      3. sqrt-unprod 1.2%

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

      4. sqr-neg 1.2%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 49.9%

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

      7. associate-*l/ 49.8%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 1.7%

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

      10. sqr-neg 1.7%

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

      11. sqrt-unprod 51.1%

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

      12. add-sqr-sqrt 51.4%

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

   10. Applied egg-rr 47.8%

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

   if 1.25e-238 < v < 9.1999999999999997e-205 or 1e-190 < 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 33.9%

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

      1. neg-mul-1 33.9%

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

   7. Simplified 33.9%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.25 \cdot 10^{-238} \lor \neg \left(v \leq 9.2 \cdot 10^{-205}\right) \land v \leq 10^{-190}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.9e-15) (- (* m (/ m v)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-15) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.9d-15) then
        tmp = (m * (m / v)) - m
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-15) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.9e-15:
		tmp = (m * (m / v)) - m
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.9e-15)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	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 <= 2.9e-15)
		tmp = (m * (m / v)) - m;
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.9e-15], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $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 < 2.90000000000000019e-15

   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* 99.7%

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

      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 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. neg-mul-1 99.8%

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

      6. unsub-neg 99.8%

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

      7. associate-/l* 85.9%

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

      8. unpow2 85.9%

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

   7. Simplified 85.9%

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

      1. unpow2 85.9%

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

      2. add-sqr-sqrt 85.7%

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

      3. sqrt-unprod 49.3%

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

      4. sqr-neg 49.3%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 53.0%

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

      7. associate-*l/ 52.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 49.9%

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

      10. sqr-neg 49.9%

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

      11. sqrt-unprod 99.5%

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

      12. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.8%

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

   if 2.90000000000000019e-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. 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 v around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.8%

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

      3. *-lft-identity 99.8%

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

      4. associate-*l/ 99.9%

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

      5. associate-/r/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-rgt1-in 99.9%

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

      10. *-rgt-identity 99.9%

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

      11. sub-neg 99.9%

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

   9. Simplified 99.9%

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

       1. associate-/r/ 99.9%

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

       2. unpow2 99.9%

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

       3. associate-*r* 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (m * (m / v)) - m
	else:
		tmp = m * ((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(Float64(m / v) * Float64(-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 * ((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[(N[(m / v), $MachinePrecision] * (-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. 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* 99.7%

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

      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 96.4%

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

      1. sub-neg 96.4%

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

      2. metadata-eval 96.4%

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

      3. distribute-lft-in 96.5%

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

      4. *-commutative 96.5%

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

      5. neg-mul-1 96.5%

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

      6. unsub-neg 96.5%

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

      7. associate-/l* 83.4%

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

      8. unpow2 83.4%

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

   7. Simplified 83.4%

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

      1. unpow2 83.4%

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

      2. add-sqr-sqrt 83.2%

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

      3. sqrt-unprod 47.7%

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

      4. sqr-neg 47.7%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 49.8%

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

      7. associate-*l/ 49.7%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 48.2%

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

      10. sqr-neg 48.2%

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

      11. sqrt-unprod 96.2%

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

      12. add-sqr-sqrt 96.5%

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

   9. Applied egg-rr 96.5%

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

   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 v around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-*l/ 99.9%

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

      5. associate-/r/ 99.9%

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

      6. associate-*r/ 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-rgt1-in 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. sub-neg 100.0%

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

   9. Simplified 100.0%

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

       1. associate-/r/ 99.9%

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

       2. unpow2 99.9%

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

       3. associate-*r* 99.9%

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

   11. Applied egg-rr 99.9%

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

   12. Taylor expanded in m around inf 98.8%

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

       1. neg-mul-1 98.8%

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

       2. distribute-neg-frac2 98.8%

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

   14. Simplified 98.8%

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

4. Final simplification 97.7%

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

Alternative 7: 87.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (m * (m / v)) - m
	else:
		tmp = (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(Float64(m / v) * Float64(-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 / 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[(N[(m / v), $MachinePrecision] * (-m)), $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. 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* 99.7%

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

      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 96.4%

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

      1. sub-neg 96.4%

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

      2. metadata-eval 96.4%

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

      3. distribute-lft-in 96.5%

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

      4. *-commutative 96.5%

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

      5. neg-mul-1 96.5%

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

      6. unsub-neg 96.5%

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

      7. associate-/l* 83.4%

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

      8. unpow2 83.4%

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

   7. Simplified 83.4%

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

      1. unpow2 83.4%

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

      2. add-sqr-sqrt 83.2%

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

      3. sqrt-unprod 47.7%

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

      4. sqr-neg 47.7%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 49.8%

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

      7. associate-*l/ 49.7%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 48.2%

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

      10. sqr-neg 48.2%

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

      11. sqrt-unprod 96.2%

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

      12. add-sqr-sqrt 96.5%

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

   9. Applied egg-rr 96.5%

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

   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%

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

      1. sub-neg 0.1%

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

      2. metadata-eval 0.1%

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

      3. distribute-lft-in 0.1%

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

      4. *-commutative 0.1%

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

      5. neg-mul-1 0.1%

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

      6. unsub-neg 0.1%

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

      7. associate-/l* 0.1%

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

      8. unpow2 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in m around inf 0.1%

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

      1. *-un-lft-identity 0.1%

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

      2. add-sqr-sqrt 0.1%

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

      3. sqrt-unprod 0.1%

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

      4. sqr-neg 0.1%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 82.5%

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

      7. neg-mul-1 82.5%

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

      8. times-frac 82.5%

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

      9. metadata-eval 82.5%

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

      10. unpow2 82.5%

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

      11. add-sqr-sqrt 82.5%

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

      12. sqrt-unprod 83.7%

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

      13. sqr-neg 83.7%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.1%

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

      16. associate-*r/ 0.1%

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

      17. *-commutative 0.1%

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

      18. *-commutative 0.1%

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

      19. associate-*l* 0.1%

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

   10. Applied egg-rr 82.5%

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

4. Final simplification 88.8%

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

Alternative 8: 87.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.0d0) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = (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 / v) * -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
	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 / v) * 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;
	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 / v), $MachinePrecision] * (-m)), $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. 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* 99.7%

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

      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 96.4%

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

   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%

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

      1. sub-neg 0.1%

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

      2. metadata-eval 0.1%

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

      3. distribute-lft-in 0.1%

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

      4. *-commutative 0.1%

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

      5. neg-mul-1 0.1%

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

      6. unsub-neg 0.1%

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

      7. associate-/l* 0.1%

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

      8. unpow2 0.1%

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

   7. Simplified 0.1%

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

   8. Taylor expanded in m around inf 0.1%

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

      1. *-un-lft-identity 0.1%

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

      2. add-sqr-sqrt 0.1%

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

      3. sqrt-unprod 0.1%

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

      4. sqr-neg 0.1%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 82.5%

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

      7. neg-mul-1 82.5%

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

      8. times-frac 82.5%

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

      9. metadata-eval 82.5%

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

      10. unpow2 82.5%

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

      11. add-sqr-sqrt 82.5%

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

      12. sqrt-unprod 83.7%

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

      13. sqr-neg 83.7%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.1%

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

      16. associate-*r/ 0.1%

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

      17. *-commutative 0.1%

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

      18. *-commutative 0.1%

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

      19. associate-*l* 0.1%

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

   10. Applied egg-rr 82.5%

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

4. Final simplification 88.8%

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

Alternative 9: 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. 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 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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 11: 27.0% 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 25.8%

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

   1. neg-mul-1 25.8%

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

7. Simplified 25.8%

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

Reproduce

herbie shell --seed 2024089 
(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))