a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 9.4s

Alternatives: 12

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: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[m_, v_] := N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision] * m + (-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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. distribute-rgt-in 99.8%

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

   2. fma-define 99.9%

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

   3. associate-*r/ 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-*r/ 99.9%

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

   6. neg-mul-1 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-148}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 - \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7.2e-148)
   (- m)
   (if (<= m 1.0) (* m (/ m v)) (* m (- -1.0 (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.2e-148) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = m * (-1.0 - (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 7.2d-148) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m * (m / v)
    else
        tmp = m * ((-1.0d0) - (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7.2e-148) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = m * (-1.0 - (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.2e-148:
		tmp = -m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = m * (-1.0 - (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.2e-148)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(m * Float64(-1.0 - Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7.2e-148)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = m * (-1.0 - (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.2e-148], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(-1.0 - N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 7.1999999999999997e-148

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-rgt-in 99.9%

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

      2. fma-define 99.9%

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

      3. associate-*r/ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r/ 99.9%

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

      6. neg-mul-1 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 70.1%

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

      1. neg-mul-1 70.1%

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

   9. Simplified 70.1%

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

   if 7.1999999999999997e-148 < m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in m around 0 96.5%

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

   6. Taylor expanded in m around inf 76.8%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 0.1%

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

      1. fma-define 0.1%

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

      2. metadata-eval 0.1%

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

      3. associate-/r* 0.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 80.3%

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

      6. mul-1-neg 80.3%

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

      7. mul-1-neg 80.3%

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

      8. sqr-neg 80.3%

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

      9. sqrt-unprod 79.0%

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

      10. add-sqr-sqrt 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. fma-undefine 79.0%

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

      2. associate-*r/ 79.0%

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

      3. *-commutative 79.0%

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

      4. mul-1-neg 79.0%

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

      5. distribute-neg-frac 79.0%

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

      6. +-commutative 79.0%

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

      7. sub-neg 79.0%

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

   9. Simplified 79.0%

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

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-20}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.65e-20) (- (* m (/ m v)) m) (* m (/ (* m (- 1.0 m)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-20) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * ((m * (1.0 - m)) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.65d-20) then
        tmp = (m * (m / v)) - m
    else
        tmp = m * ((m * (1.0d0 - m)) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-20) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * ((m * (1.0 - m)) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.65e-20:
		tmp = (m * (m / v)) - m
	else:
		tmp = m * ((m * (1.0 - m)) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.65e-20)
		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 <= 2.65e-20)
		tmp = (m * (m / v)) - m;
	else
		tmp = m * ((m * (1.0 - m)) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.65e-20], 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 < 2.6500000000000001e-20

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 99.7%

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

   6. Taylor expanded in v around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. unsub-neg 99.8%

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

   8. Simplified 99.8%

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

      1. div-sub 99.8%

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

      2. div-inv 99.7%

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

      3. *-inverses 99.7%

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

      4. associate-/l* 99.8%

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

      5. rgt-mult-inverse 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. distribute-rgt-in 99.8%

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

      9. rgt-mult-inverse 99.9%

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

      10. *-rgt-identity 99.9%

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

      11. neg-mul-1 99.9%

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

   10. Applied egg-rr 99.9%

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

   if 2.6500000000000001e-20 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fma-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in v around 0 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-20}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-20}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.65e-20) (- (* m (/ m v)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-20) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.65d-20) then
        tmp = (m * (m / v)) - m
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-20) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.65e-20:
		tmp = (m * (m / v)) - m
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.65e-20)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.65e-20)
		tmp = (m * (m / v)) - m;
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.65e-20], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.6500000000000001e-20

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 99.7%

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

   6. Taylor expanded in v around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. unsub-neg 99.8%

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

   8. Simplified 99.8%

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

      1. div-sub 99.8%

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

      2. div-inv 99.7%

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

      3. *-inverses 99.7%

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

      4. associate-/l* 99.8%

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

      5. rgt-mult-inverse 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. distribute-rgt-in 99.8%

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

      9. rgt-mult-inverse 99.9%

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

      10. *-rgt-identity 99.9%

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

      11. neg-mul-1 99.9%

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

   10. Applied egg-rr 99.9%

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

   if 2.6500000000000001e-20 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fma-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in v around inf 99.9%

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

      1. sub-neg 99.9%

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

      2. associate-/l* 99.9%

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

      3. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in v around 0 99.4%

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

       1. associate-*r/ 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-20}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
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. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 98.2%

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

   6. Taylor expanded in v around 0 98.2%

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

      1. neg-mul-1 98.2%

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

      2. unsub-neg 98.2%

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

   8. Simplified 98.2%

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

      1. div-sub 98.2%

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

      2. div-inv 98.2%

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

      3. *-inverses 98.2%

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

      4. associate-/l* 98.2%

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

      5. rgt-mult-inverse 98.2%

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

      6. sub-neg 98.2%

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

      7. metadata-eval 98.2%

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

      8. distribute-rgt-in 98.2%

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

      9. rgt-mult-inverse 98.3%

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

      10. *-rgt-identity 98.3%

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

      11. neg-mul-1 98.3%

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

   10. Applied egg-rr 98.3%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 0.1%

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

      1. fma-define 0.1%

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

      2. metadata-eval 0.1%

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

      3. associate-/r* 0.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 80.3%

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

      6. mul-1-neg 80.3%

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

      7. mul-1-neg 80.3%

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

      8. sqr-neg 80.3%

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

      9. sqrt-unprod 79.0%

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

      10. add-sqr-sqrt 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. fma-undefine 79.0%

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

      2. associate-*r/ 79.0%

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

      3. *-commutative 79.0%

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

      4. mul-1-neg 79.0%

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

      5. distribute-neg-frac 79.0%

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

      6. +-commutative 79.0%

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

      7. sub-neg 79.0%

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

   9. Simplified 79.0%

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

4. Final simplification 87.8%

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

Alternative 6: 86.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = m * ((m / v) + -1.0)
	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(Float64(m / v) + -1.0));
	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) + -1.0);
	else
		tmp = m * (-1.0 - (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[(-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 98.2%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 0.1%

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

      1. fma-define 0.1%

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

      2. metadata-eval 0.1%

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

      3. associate-/r* 0.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 80.3%

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

      6. mul-1-neg 80.3%

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

      7. mul-1-neg 80.3%

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

      8. sqr-neg 80.3%

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

      9. sqrt-unprod 79.0%

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

      10. add-sqr-sqrt 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. fma-undefine 79.0%

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

      2. associate-*r/ 79.0%

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

      3. *-commutative 79.0%

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

      4. mul-1-neg 79.0%

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

      5. distribute-neg-frac 79.0%

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

      6. +-commutative 79.0%

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

      7. sub-neg 79.0%

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

   9. Simplified 79.0%

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

4. Final simplification 87.8%

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

Alternative 7: 86.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = m * ((m - v) / 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(Float64(m - v) / 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 * ((m - v) / 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[(N[(m - v), $MachinePrecision] / v), $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. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 98.2%

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

   6. Taylor expanded in v around 0 98.2%

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

      1. neg-mul-1 98.2%

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

      2. unsub-neg 98.2%

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

   8. Simplified 98.2%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 0.1%

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

      1. fma-define 0.1%

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

      2. metadata-eval 0.1%

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

      3. associate-/r* 0.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 80.3%

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

      6. mul-1-neg 80.3%

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

      7. mul-1-neg 80.3%

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

      8. sqr-neg 80.3%

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

      9. sqrt-unprod 79.0%

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

      10. add-sqr-sqrt 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. fma-undefine 79.0%

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

      2. associate-*r/ 79.0%

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

      3. *-commutative 79.0%

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

      4. mul-1-neg 79.0%

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

      5. distribute-neg-frac 79.0%

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

      6. +-commutative 79.0%

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

      7. sub-neg 79.0%

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

   9. Simplified 79.0%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Final simplification 99.9%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

Alternative 10: 38.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.12 \cdot 10^{-155}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 1.12e-155) (* m (/ m v)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 1.12e-155) {
		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 <= 1.12d-155) 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 <= 1.12e-155) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 1.12e-155:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 1.12e-155)
		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 <= 1.12e-155)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 1.12e-155], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 1.1200000000000001e-155

   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 50.3%

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

   6. Taylor expanded in m around inf 38.8%

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

   if 1.1200000000000001e-155 < v

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. fma-define 99.9%

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

      3. associate-*r/ 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r/ 99.9%

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

      6. neg-mul-1 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 34.0%

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

      1. neg-mul-1 34.0%

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

   9. Simplified 34.0%

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

Alternative 11: 27.3% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. distribute-rgt-in 99.8%

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

   2. fma-define 99.9%

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

   3. associate-*r/ 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-*r/ 99.9%

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

   6. neg-mul-1 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in m around 0 23.0%

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

   1. neg-mul-1 23.0%

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

9. Simplified 23.0%

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

Alternative 12: 3.0% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (m v) :precision binary64 m)

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return m

Julia

function code(m, v)
	return m
end

MATLAB

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

Wolfram

code[m_, v_] := m

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-/l* 99.9%

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

   4. fma-neg 99.9%

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

   5. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in m around 0 23.0%

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

   1. *-commutative 23.0%

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

   2. neg-mul-1 23.0%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 2.8%

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

   5. sqr-neg 2.8%

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

   6. sqrt-unprod 2.7%

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

   7. add-sqr-sqrt 2.7%

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

   8. /-rgt-identity 2.7%

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

7. Applied egg-rr 2.7%

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

8. Taylor expanded in m around 0 2.7%

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

Reproduce

herbie shell --seed 2024135 
(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))