a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.0%

Time: 8.7s

Alternatives: 11

Speedup: 1.0×

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

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.2 \cdot 10^{-49}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.2e-49) (- (* m (/ m v)) m) (/ (* (- 1.0 m) (* m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.2e-49) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = ((1.0 - m) * (m * m)) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.2d-49) then
        tmp = (m * (m / v)) - m
    else
        tmp = ((1.0d0 - m) * (m * m)) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.2e-49) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = ((1.0 - m) * (m * m)) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.2e-49:
		tmp = (m * (m / v)) - m
	else:
		tmp = ((1.0 - m) * (m * m)) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.2e-49)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(m * m)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.2e-49)
		tmp = (m * (m / v)) - m;
	else
		tmp = ((1.0 - m) * (m * m)) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.2e-49], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.1999999999999999e-49

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 99.8%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 92.3%

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

      3. sqr-neg 92.3%

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

      4. distribute-frac-neg 92.3%

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

      5. distribute-frac-neg 92.3%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 53.5%

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

      8. distribute-frac-neg 53.5%

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

   5. Applied egg-rr 53.5%

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

      1. *-commutative 53.5%

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

      2. sub-neg 53.5%

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

      3. metadata-eval 53.5%

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

      4. distribute-rgt-in 53.5%

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

      5. distribute-neg-frac2 53.5%

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

      6. div-inv 53.5%

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

      7. metadata-eval 53.5%

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

      8. frac-2neg 53.5%

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

      9. *-commutative 53.5%

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

      10. frac-2neg 53.5%

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

      11. metadata-eval 53.5%

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

      12. div-inv 53.5%

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

      13. distribute-neg-frac2 53.5%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 92.3%

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

      16. sqr-neg 92.3%

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

      17. sqrt-unprod 99.6%

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

      18. add-sqr-sqrt 99.8%

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

      19. neg-mul-1 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 2.1999999999999999e-49 < m

   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 v around 0 99.8%

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

      1. unpow2 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.2 \cdot 10^{-49}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-52}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5.5e-52) (- (* m (/ m v)) m) (* m (/ (* m (- 1.0 m)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.5e-52) {
		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 <= 5.5d-52) 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 <= 5.5e-52) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * ((m * (1.0 - m)) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.5e-52:
		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 <= 5.5e-52)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(m * Float64(Float64(m * Float64(1.0 - m)) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.5e-52)
		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, 5.5e-52], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.5e-52

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 99.8%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 92.3%

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

      3. sqr-neg 92.3%

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

      4. distribute-frac-neg 92.3%

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

      5. distribute-frac-neg 92.3%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 54.0%

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

      8. distribute-frac-neg 54.0%

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

   5. Applied egg-rr 54.0%

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

      1. *-commutative 54.0%

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

      2. sub-neg 54.0%

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

      3. metadata-eval 54.0%

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

      4. distribute-rgt-in 54.0%

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

      5. distribute-neg-frac2 54.0%

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

      6. div-inv 54.0%

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

      7. metadata-eval 54.0%

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

      8. frac-2neg 54.0%

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

      9. *-commutative 54.0%

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

      10. frac-2neg 54.0%

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

      11. metadata-eval 54.0%

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

      12. div-inv 54.0%

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

      13. distribute-neg-frac2 54.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 92.3%

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

      16. sqr-neg 92.3%

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

      17. sqrt-unprod 99.6%

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

      18. add-sqr-sqrt 99.8%

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

      19. neg-mul-1 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 5.5e-52 < m

   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 v around 0 99.8%

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

      1. unpow2 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-*l* 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. times-frac 99.8%

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

      4. /-rgt-identity 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-52}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.2 \cdot 10^{-41}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.2e-41) (- (* m (/ m v)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.2e-41) {
		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 <= 6.2d-41) 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 <= 6.2e-41) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.2e-41:
		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 <= 6.2e-41)
		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 <= 6.2e-41)
		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, 6.2e-41], 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 < 6.20000000000000001e-41

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 99.8%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 91.1%

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

      3. sqr-neg 91.1%

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

      4. distribute-frac-neg 91.1%

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

      5. distribute-frac-neg 91.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 50.1%

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

      8. distribute-frac-neg 50.1%

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

   5. Applied egg-rr 50.1%

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

      1. *-commutative 50.1%

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

      2. sub-neg 50.1%

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

      3. metadata-eval 50.1%

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

      4. distribute-rgt-in 50.1%

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

      5. distribute-neg-frac2 50.1%

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

      6. div-inv 50.1%

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

      7. metadata-eval 50.1%

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

      8. frac-2neg 50.1%

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

      9. *-commutative 50.1%

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

      10. frac-2neg 50.1%

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

      11. metadata-eval 50.1%

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

      12. div-inv 50.1%

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

      13. distribute-neg-frac2 50.1%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 91.1%

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

      16. sqr-neg 91.1%

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

      17. sqrt-unprod 99.6%

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

      18. add-sqr-sqrt 99.8%

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

      19. neg-mul-1 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 6.20000000000000001e-41 < m

   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 v around 0 99.8%

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

      1. unpow2 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-/l* 99.8%

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

      2. associate-*l* 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.2 \cdot 10^{-41}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0) (- (* m (/ m v)) m) (/ (* 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 * -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 * -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 * -m)) / v;
	}
	return tmp;
}

Python

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 95.6%

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

      1. add-sqr-sqrt 95.4%

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

      2. sqrt-unprod 85.2%

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

      3. sqr-neg 85.2%

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

      4. distribute-frac-neg 85.2%

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

      5. distribute-frac-neg 85.2%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 43.5%

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

      8. distribute-frac-neg 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. *-commutative 43.5%

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

      2. sub-neg 43.5%

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

      3. metadata-eval 43.5%

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

      4. distribute-rgt-in 43.5%

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

      5. distribute-neg-frac2 43.5%

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

      6. div-inv 43.5%

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

      7. metadata-eval 43.5%

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

      8. frac-2neg 43.5%

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

      9. *-commutative 43.5%

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

      10. frac-2neg 43.5%

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

      11. metadata-eval 43.5%

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

      12. div-inv 43.5%

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

      13. distribute-neg-frac2 43.5%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 85.2%

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

      16. sqr-neg 85.2%

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

      17. sqrt-unprod 95.4%

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

      18. add-sqr-sqrt 95.6%

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

      19. neg-mul-1 95.6%

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

   7. Applied egg-rr 95.6%

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

   if 1 < m

   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 v around 0 99.9%

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

      1. unpow2 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in m around inf 96.6%

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

      1. neg-mul-1 96.6%

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

   10. Simplified 96.6%

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

4. Final simplification 96.1%

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

Alternative 5: 86.7% accurate, 0.9× speedup

Math

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

FPCore

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

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));
	}
	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))
    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));
	}
	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))
	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 / 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 * (-1.0 - (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[(-1.0 - N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 95.6%

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

      1. add-sqr-sqrt 95.4%

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

      2. sqrt-unprod 85.2%

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

      3. sqr-neg 85.2%

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

      4. distribute-frac-neg 85.2%

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

      5. distribute-frac-neg 85.2%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 43.5%

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

      8. distribute-frac-neg 43.5%

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

   5. Applied egg-rr 43.5%

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

      1. *-commutative 43.5%

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

      2. sub-neg 43.5%

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

      3. metadata-eval 43.5%

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

      4. distribute-rgt-in 43.5%

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

      5. distribute-neg-frac2 43.5%

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

      6. div-inv 43.5%

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

      7. metadata-eval 43.5%

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

      8. frac-2neg 43.5%

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

      9. *-commutative 43.5%

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

      10. frac-2neg 43.5%

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

      11. metadata-eval 43.5%

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

      12. div-inv 43.5%

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

      13. distribute-neg-frac2 43.5%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 85.2%

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

      16. sqr-neg 85.2%

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

      17. sqrt-unprod 95.4%

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

      18. add-sqr-sqrt 95.6%

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

      19. neg-mul-1 95.6%

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

   7. Applied egg-rr 95.6%

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

   if 1 < m

   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 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. distribute-frac-neg 0.1%

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

      5. distribute-frac-neg 0.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 72.8%

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

      8. distribute-frac-neg 72.8%

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

   5. Applied egg-rr 72.8%

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

   6. Taylor expanded in m around 0 72.8%

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

      1. neg-mul-1 72.8%

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

      2. neg-sub0 72.8%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{m}{v}\right)} - 1\right) \cdot m \]

      3. associate--r+ 72.8%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{m}{v} + 1\right)\right)} \cdot m \]

      4. +-commutative 72.8%

         \[\leadsto \left(0 - \color{blue}{\left(1 + \frac{m}{v}\right)}\right) \cdot m \]

      5. associate--r+ 72.8%

         \[\leadsto \color{blue}{\left(\left(0 - 1\right) - \frac{m}{v}\right)} \cdot m \]

      6. metadata-eval 72.8%

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

   8. Simplified 72.8%

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

4. Final simplification 84.0%

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

Alternative 6: 86.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 95.6%

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

   if 1 < m

   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 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. distribute-frac-neg 0.1%

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

      5. distribute-frac-neg 0.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 72.8%

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

      8. distribute-frac-neg 72.8%

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

   5. Applied egg-rr 72.8%

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

   6. Taylor expanded in m around 0 72.8%

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

      1. neg-mul-1 72.8%

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

      2. neg-sub0 72.8%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{m}{v}\right)} - 1\right) \cdot m \]

      3. associate--r+ 72.8%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{m}{v} + 1\right)\right)} \cdot m \]

      4. +-commutative 72.8%

         \[\leadsto \left(0 - \color{blue}{\left(1 + \frac{m}{v}\right)}\right) \cdot m \]

      5. associate--r+ 72.8%

         \[\leadsto \color{blue}{\left(\left(0 - 1\right) - \frac{m}{v}\right)} \cdot m \]

      6. metadata-eval 72.8%

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

   8. Simplified 72.8%

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

4. Final simplification 84.0%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(m * N[(N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $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. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. *-un-lft-identity 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 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. Taylor expanded in m around 0 99.7%

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

   1. neg-mul-1 99.7%

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

   2. +-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. div-sub 99.8%

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

   5. associate-*r/ 99.8%

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

   6. associate-*l/ 99.8%

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

   7. associate-/r/ 99.8%

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

7. Simplified 99.8%

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

Alternative 9: 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 10: 62.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return 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 / v))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in m around 0 47.1%

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

   1. add-sqr-sqrt 47.0%

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

   2. sqrt-unprod 42.0%

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

   3. sqr-neg 42.0%

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

   4. distribute-frac-neg 42.0%

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

   5. distribute-frac-neg 42.0%

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

   6. sqrt-unprod 0.0%

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

   7. add-sqr-sqrt 58.4%

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

   8. distribute-frac-neg 58.4%

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

5. Applied egg-rr 58.4%

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

6. Taylor expanded in m around 0 58.4%

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

   1. neg-mul-1 58.4%

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

   2. neg-sub0 58.4%

      \[\leadsto \left(\color{blue}{\left(0 - \frac{m}{v}\right)} - 1\right) \cdot m \]

   3. associate--r+ 58.4%

      \[\leadsto \color{blue}{\left(0 - \left(\frac{m}{v} + 1\right)\right)} \cdot m \]

   4. +-commutative 58.4%

      \[\leadsto \left(0 - \color{blue}{\left(1 + \frac{m}{v}\right)}\right) \cdot m \]

   5. associate--r+ 58.4%

      \[\leadsto \color{blue}{\left(\left(0 - 1\right) - \frac{m}{v}\right)} \cdot m \]

   6. metadata-eval 58.4%

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

8. Simplified 58.4%

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

9. Final simplification 58.4%

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

Alternative 11: 27.4% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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 99.7%

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

   1. neg-mul-1 99.7%

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

   2. +-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. div-sub 99.8%

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

   5. associate-*r/ 99.8%

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

   6. associate-*l/ 99.8%

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

   7. associate-/r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in m around 0 24.2%

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

   1. mul-1-neg 24.2%

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

10. Simplified 24.2%

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

Reproduce

herbie shell --seed 2024156 
(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))