a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 10.0s

Alternatives: 13

Speedup: 1.0×

• 41.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. distribute-rgt-in 99.9%

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

   2. associate-/r/ 99.9%

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

   3. associate-*l* 99.9%

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

   4. *-commutative 99.9%

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

   5. neg-mul-1 99.9%

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

   6. fma-def 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 71.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \frac{m}{v}\\ \mathbf{if}\;m \leq 3.2 \cdot 10^{-131}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.9 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 3.4 \cdot 10^{-64}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{-v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (/ m v))))
   (if (<= m 3.2e-131)
     (- m)
     (if (<= m 1.9e-96)
       t_0
       (if (<= m 3.4e-64) (- m) (if (<= m 1.0) t_0 (* m (/ m (- v)))))))))

C

double code(double m, double v) {
	double t_0 = m * (m / v);
	double tmp;
	if (m <= 3.2e-131) {
		tmp = -m;
	} else if (m <= 1.9e-96) {
		tmp = t_0;
	} else if (m <= 3.4e-64) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = t_0;
	} else {
		tmp = m * (m / -v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (m / v)
    if (m <= 3.2d-131) then
        tmp = -m
    else if (m <= 1.9d-96) then
        tmp = t_0
    else if (m <= 3.4d-64) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = t_0
    else
        tmp = m * (m / -v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = m * (m / v);
	double tmp;
	if (m <= 3.2e-131) {
		tmp = -m;
	} else if (m <= 1.9e-96) {
		tmp = t_0;
	} else if (m <= 3.4e-64) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = t_0;
	} else {
		tmp = m * (m / -v);
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (m / v)
	tmp = 0
	if m <= 3.2e-131:
		tmp = -m
	elif m <= 1.9e-96:
		tmp = t_0
	elif m <= 3.4e-64:
		tmp = -m
	elif m <= 1.0:
		tmp = t_0
	else:
		tmp = m * (m / -v)
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(m / v))
	tmp = 0.0
	if (m <= 3.2e-131)
		tmp = Float64(-m);
	elseif (m <= 1.9e-96)
		tmp = t_0;
	elseif (m <= 3.4e-64)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(m * Float64(m / Float64(-v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (m / v);
	tmp = 0.0;
	if (m <= 3.2e-131)
		tmp = -m;
	elseif (m <= 1.9e-96)
		tmp = t_0;
	elseif (m <= 3.4e-64)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = t_0;
	else
		tmp = m * (m / -v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 3.2e-131], (-m), If[LessEqual[m, 1.9e-96], t$95$0, If[LessEqual[m, 3.4e-64], (-m), If[LessEqual[m, 1.0], t$95$0, N[(m * N[(m / (-v)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if m < 3.2e-131 or 1.9e-96 < m < 3.40000000000000012e-64

   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%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 72.8%

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

      1. neg-mul-1 72.8%

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

   6. Simplified 72.8%

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

   if 3.2e-131 < m < 1.9e-96 or 3.40000000000000012e-64 < 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%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in v around 0 79.8%

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

      1. associate-/l* 79.9%

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

      2. unpow2 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in m around 0 72.1%

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

      1. associate-*l/ 72.3%

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

   9. Applied egg-rr 72.3%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 76.5%

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

      6. neg-mul-1 76.5%

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

      7. times-frac 76.5%

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

   9. Applied egg-rr 76.5%

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

   10. Taylor expanded in m around 0 76.5%

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

       1. metadata-eval 76.5%

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

       2. unpow2 76.5%

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

       3. associate-/l* 76.5%

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

       4. associate-/r/ 76.5%

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

       5. associate-/r/ 76.5%

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

       6. times-frac 76.5%

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

       7. neg-mul-1 76.5%

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

       8. associate-/r/ 76.5%

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

       9. *-commutative 76.5%

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

       10. associate-*r/ 76.5%

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

       11. *-rgt-identity 76.5%

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

       12. associate-*l/ 76.5%

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

       13. *-commutative 76.5%

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

   12. Simplified 76.5%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{-131}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.9 \cdot 10^{-96}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 3.4 \cdot 10^{-64}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{-v}\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-9}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7e-9) (- (* m (/ m v)) m) (* m (/ (- 1.0 m) (/ v m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7e-9) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * ((1.0 - m) / (v / m));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 7d-9) then
        tmp = (m * (m / v)) - m
    else
        tmp = m * ((1.0d0 - m) / (v / m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7e-9) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * ((1.0 - m) / (v / m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7e-9:
		tmp = (m * (m / v)) - m
	else:
		tmp = m * ((1.0 - m) / (v / m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7e-9)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(m * Float64(Float64(1.0 - m) / Float64(v / m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7e-9)
		tmp = (m * (m / v)) - m;
	else
		tmp = m * ((1.0 - m) / (v / m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7e-9], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(N[(1.0 - m), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.9999999999999998e-9

   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%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 84.1%

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

      1. neg-mul-1 84.1%

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

      2. +-commutative 84.1%

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

      3. unsub-neg 84.1%

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

      4. unpow2 84.1%

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

      5. associate-/l* 98.9%

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

   6. Simplified 98.9%

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

      1. associate-/r/ 99.0%

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

   8. Applied egg-rr 99.0%

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

   if 6.9999999999999998e-9 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-*l/ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r/ 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r/ 99.9%

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

      7. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-9}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \frac{m}{v}\\ \mathbf{if}\;m \leq 7 \cdot 10^{-9}:\\ \;\;\;\;t_0 - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (/ m v)))) (if (<= m 7e-9) (- t_0 m) (* (- 1.0 m) t_0))))

C

double code(double m, double v) {
	double t_0 = m * (m / v);
	double tmp;
	if (m <= 7e-9) {
		tmp = t_0 - m;
	} else {
		tmp = (1.0 - m) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (m / v)
    if (m <= 7d-9) then
        tmp = t_0 - m
    else
        tmp = (1.0d0 - m) * t_0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = m * (m / v);
	double tmp;
	if (m <= 7e-9) {
		tmp = t_0 - m;
	} else {
		tmp = (1.0 - m) * t_0;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (m / v)
	tmp = 0
	if m <= 7e-9:
		tmp = t_0 - m
	else:
		tmp = (1.0 - m) * t_0
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(m / v))
	tmp = 0.0
	if (m <= 7e-9)
		tmp = Float64(t_0 - m);
	else
		tmp = Float64(Float64(1.0 - m) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (m / v);
	tmp = 0.0;
	if (m <= 7e-9)
		tmp = t_0 - m;
	else
		tmp = (1.0 - m) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 7e-9], N[(t$95$0 - m), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 6.9999999999999998e-9

   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%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 84.1%

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

      1. neg-mul-1 84.1%

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

      2. +-commutative 84.1%

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

      3. unsub-neg 84.1%

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

      4. unpow2 84.1%

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

      5. associate-/l* 98.9%

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

   6. Simplified 98.9%

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

      1. associate-/r/ 99.0%

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

   8. Applied egg-rr 99.0%

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

   if 6.9999999999999998e-9 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. associate-/r/ 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

      5. neg-mul-1 99.9%

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

      6. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in v around 0 99.9%

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

      1. unpow2 99.9%

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

      2. associate-/l* 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-*r/ 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-9}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-9}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7e-9) (- (* m (/ m v)) m) (/ (* m m) (/ v (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7e-9) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = (m * m) / (v / (1.0 - m));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 7d-9) then
        tmp = (m * (m / v)) - m
    else
        tmp = (m * m) / (v / (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7e-9) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = (m * m) / (v / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7e-9:
		tmp = (m * (m / v)) - m
	else:
		tmp = (m * m) / (v / (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7e-9)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(Float64(m * m) / Float64(v / Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7e-9)
		tmp = (m * (m / v)) - m;
	else
		tmp = (m * m) / (v / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7e-9], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m * m), $MachinePrecision] / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.9999999999999998e-9

   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%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 84.1%

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

      1. neg-mul-1 84.1%

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

      2. +-commutative 84.1%

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

      3. unsub-neg 84.1%

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

      4. unpow2 84.1%

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

      5. associate-/l* 98.9%

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

   6. Simplified 98.9%

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

      1. associate-/r/ 99.0%

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

   8. Applied egg-rr 99.0%

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

   if 6.9999999999999998e-9 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-9}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 6: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.8%

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

   2. associate-/r/ 99.8%

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

   3. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 8: 97.6% accurate, 1.1× speedup

Math

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

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(Float64(m * 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[((-m) * N[(N[(m * m), $MachinePrecision] / v), $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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 83.0%

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

      1. neg-mul-1 83.0%

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

      2. +-commutative 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. associate-/l* 97.5%

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

   6. Simplified 97.5%

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

      1. associate-/r/ 97.5%

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

   8. Applied egg-rr 97.5%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 95.9%

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

      1. associate-*r/ 95.9%

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

      2. neg-mul-1 95.9%

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

   9. Simplified 95.9%

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

       1. frac-2neg 95.9%

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

       2. remove-double-neg 95.9%

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

       3. associate-/r/ 95.8%

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

   11. Applied egg-rr 95.8%

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

4. Final simplification 96.8%

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

Alternative 9: 97.6% accurate, 1.1× speedup

Math

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

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(Float64(m * m) / Float64(Float64(-v) / m));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 83.0%

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

      1. neg-mul-1 83.0%

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

      2. +-commutative 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. associate-/l* 97.5%

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

   6. Simplified 97.5%

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

      1. associate-/r/ 97.5%

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

   8. Applied egg-rr 97.5%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 95.9%

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

      1. associate-*r/ 95.9%

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

      2. neg-mul-1 95.9%

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

   9. Simplified 95.9%

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

4. Final simplification 96.8%

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

Alternative 10: 87.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / (-v)), $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. Taylor expanded in m around 0 97.5%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 76.5%

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

      6. neg-mul-1 76.5%

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

      7. times-frac 76.5%

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

   9. Applied egg-rr 76.5%

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

   10. Taylor expanded in m around 0 76.5%

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

       1. metadata-eval 76.5%

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

       2. unpow2 76.5%

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

       3. associate-/l* 76.5%

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

       4. associate-/r/ 76.5%

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

       5. associate-/r/ 76.5%

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

       6. times-frac 76.5%

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

       7. neg-mul-1 76.5%

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

       8. associate-/r/ 76.5%

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

       9. *-commutative 76.5%

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

       10. associate-*r/ 76.5%

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

       11. *-rgt-identity 76.5%

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

       12. associate-*l/ 76.5%

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

       13. *-commutative 76.5%

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

   12. Simplified 76.5%

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

4. Final simplification 88.1%

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

Alternative 11: 87.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 83.0%

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

      1. neg-mul-1 83.0%

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

      2. +-commutative 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. associate-/l* 97.5%

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

   6. Simplified 97.5%

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

      1. associate-/r/ 97.5%

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

   8. Applied egg-rr 97.5%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 76.5%

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

      6. neg-mul-1 76.5%

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

      7. times-frac 76.5%

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

   9. Applied egg-rr 76.5%

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

   10. Taylor expanded in m around 0 76.5%

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

       1. metadata-eval 76.5%

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

       2. unpow2 76.5%

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

       3. associate-/l* 76.5%

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

       4. associate-/r/ 76.5%

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

       5. associate-/r/ 76.5%

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

       6. times-frac 76.5%

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

       7. neg-mul-1 76.5%

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

       8. associate-/r/ 76.5%

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

       9. *-commutative 76.5%

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

       10. associate-*r/ 76.5%

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

       11. *-rgt-identity 76.5%

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

       12. associate-*l/ 76.5%

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

       13. *-commutative 76.5%

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

   12. Simplified 76.5%

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

4. Final simplification 88.1%

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

Alternative 12: 37.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.66 \cdot 10^{-190}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 1.66e-190) (* m (/ m v)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 1.66e-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 <= 1.66d-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 <= 1.66e-190) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 1.66e-190:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 1.66e-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 <= 1.66e-190)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 1.66e-190], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 1.6600000000000001e-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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in v around 0 74.1%

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

      1. associate-/l* 74.0%

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

      2. unpow2 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in m around 0 21.5%

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

      1. associate-*l/ 38.3%

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

   9. Applied egg-rr 38.3%

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

   if 1.6600000000000001e-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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 48.8%

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

      1. neg-mul-1 48.8%

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

   6. Simplified 48.8%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.66 \cdot 10^{-190}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 13: 26.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 34.2%

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

   1. neg-mul-1 34.2%

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

6. Simplified 34.2%

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

7. Final simplification 34.2%

   \[\leadsto -m \]

Reproduce

herbie shell --seed 2023299 
(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))