a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 9.6s

Alternatives: 14

Speedup: 1.0×

• 40.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;-\frac{m \cdot m}{v}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (+ (/ (* m (- 1.0 m)) v) -1.0))))
   (if (<= t_0 -1e+23)
     (- (/ (* m m) v))
     (if (<= t_0 -2e-308) (- m) (* m (/ m v))))))

C

double code(double m, double v) {
	double t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	double tmp;
	if (t_0 <= -1e+23) {
		tmp = -((m * m) / v);
	} else if (t_0 <= -2e-308) {
		tmp = -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) :: t_0
    real(8) :: tmp
    t_0 = m * (((m * (1.0d0 - m)) / v) + (-1.0d0))
    if (t_0 <= (-1d+23)) then
        tmp = -((m * m) / v)
    else if (t_0 <= (-2d-308)) then
        tmp = -m
    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 * (1.0 - m)) / v) + -1.0);
	double tmp;
	if (t_0 <= -1e+23) {
		tmp = -((m * m) / v);
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0)
	tmp = 0
	if t_0 <= -1e+23:
		tmp = -((m * m) / v)
	elif t_0 <= -2e-308:
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0))
	tmp = 0.0
	if (t_0 <= -1e+23)
		tmp = Float64(-Float64(Float64(m * m) / v));
	elseif (t_0 <= -2e-308)
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	tmp = 0.0;
	if (t_0 <= -1e+23)
		tmp = -((m * m) / v);
	elseif (t_0 <= -2e-308)
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+23], (-N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), If[LessEqual[t$95$0, -2e-308], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -9.9999999999999992e22

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 0.1

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

   5. Simplified 0.1%

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

   6. Taylor expanded in m around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 0.1

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

   8. Simplified 0.1%

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

      1. associate-*l/ N/A

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

      2. associate-/r/ N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-inverses N/A

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

      14. frac-2neg N/A

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

      15. *-inverses N/A

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

      16. clear-num N/A

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

      17. /-rgt-identity N/A

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

      18. remove-double-neg N/A

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

      19. neg-sub0 N/A

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

      20. flip3-- N/A

          \[\leadsto \frac{m}{v} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{{0}^{3} - {m}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}}\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \frac{m}{v} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{0} - {m}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right) \]

   10. Applied egg-rr 77.5%

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

   if -9.9999999999999992e22 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(m\right)} \]

      2. lower-neg.f64 96.7

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

   5. Simplified 96.7%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.5%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.9

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

   5. Simplified 76.9%

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

   6. Taylor expanded in m around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 74.6

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

   8. Simplified 74.6%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 89.3

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

   10. Applied egg-rr 89.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -1 \cdot 10^{+23}:\\ \;\;\;\;-\frac{m \cdot m}{v}\\ \mathbf{elif}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-\mathsf{fma}\left(m, \frac{m}{v}, m\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e-308)
   (- (fma m (/ m v) m))
   (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308) {
		tmp = -fma(m, (m / v), m);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e-308)
		tmp = Float64(-fma(m, Float64(m / v), m));
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e-308], (-N[(m * N[(m / v), $MachinePrecision] + m), $MachinePrecision]), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 35.5

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

   5. Simplified 35.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 35.7

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

   7. Applied egg-rr 35.7%

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

   9. Applied egg-rr 84.3%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.5%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.9

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

   5. Simplified 76.9%

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

   6. Taylor expanded in m around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 74.6

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

   8. Simplified 74.6%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 89.3

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

   10. Applied egg-rr 89.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-\mathsf{fma}\left(m, \frac{m}{v}, m\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;v \cdot \frac{-m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e-308)
   (* v (/ (- m) v))
   (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308) {
		tmp = v * (-m / v);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((m * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-2d-308)) then
        tmp = v * (-m / v)
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308) {
		tmp = v * (-m / v);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308:
		tmp = v * (-m / v)
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e-308)
		tmp = Float64(v * Float64(Float64(-m) / v));
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308)
		tmp = v * (-m / v);
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e-308], N[(v * N[((-m) / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. unsub-neg N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. +-commutative N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.9

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

   8. Simplified 71.9%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.5%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.9

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

   5. Simplified 76.9%

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

   6. Taylor expanded in m around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 74.6

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

   8. Simplified 74.6%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 89.3

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

   10. Applied egg-rr 89.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;v \cdot \frac{-m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -2e-308) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308) {
		tmp = -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 * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-2d-308)) then
        tmp = -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 * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308:
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -2e-308)
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -2e-308)
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -2e-308], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(m\right)} \]

      2. lower-neg.f64 37.9

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

   5. Simplified 37.9%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.5%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.9

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

   5. Simplified 76.9%

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

   6. Taylor expanded in m around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 74.6

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

   8. Simplified 74.6%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 89.3

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

   10. Applied egg-rr 89.3%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.6 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.6e-13) (fma (/ m v) m (- m)) (/ (* m (- m (* m m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.6e-13) {
		tmp = fma((m / v), m, -m);
	} else {
		tmp = (m * (m - (m * m))) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.6e-13)
		tmp = fma(Float64(m / v), m, Float64(-m));
	else
		tmp = Float64(Float64(m * Float64(m - Float64(m * m))) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.6e-13], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision], N[(N[(m * N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.6000000000000001e-13

   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

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 91.0

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

   5. Simplified 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if 6.6000000000000001e-13 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\frac{1}{m \cdot v} \cdot {m}^{3} - \frac{1}{v} \cdot {m}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{m \cdot v}} - \frac{1}{v} \cdot {m}^{3} \]

      3. cube-mult N/A

         \[\leadsto \color{blue}{\left(m \cdot \left(m \cdot m\right)\right)} \cdot \frac{1}{m \cdot v} - \frac{1}{v} \cdot {m}^{3} \]

      4. unpow2 N/A

         \[\leadsto \left(m \cdot \color{blue}{{m}^{2}}\right) \cdot \frac{1}{m \cdot v} - \frac{1}{v} \cdot {m}^{3} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{m \cdot \left({m}^{2} \cdot \frac{1}{m \cdot v}\right)} - \frac{1}{v} \cdot {m}^{3} \]

      6. associate-/r* N/A

         \[\leadsto m \cdot \left({m}^{2} \cdot \color{blue}{\frac{\frac{1}{m}}{v}}\right) - \frac{1}{v} \cdot {m}^{3} \]

      7. associate-*r/ N/A

         \[\leadsto m \cdot \color{blue}{\frac{{m}^{2} \cdot \frac{1}{m}}{v}} - \frac{1}{v} \cdot {m}^{3} \]

      8. unpow2 N/A

         \[\leadsto m \cdot \frac{\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{m}}{v} - \frac{1}{v} \cdot {m}^{3} \]

      9. associate-*l* N/A

         \[\leadsto m \cdot \frac{\color{blue}{m \cdot \left(m \cdot \frac{1}{m}\right)}}{v} - \frac{1}{v} \cdot {m}^{3} \]

      10. rgt-mult-inverse N/A

          \[\leadsto m \cdot \frac{m \cdot \color{blue}{1}}{v} - \frac{1}{v} \cdot {m}^{3} \]

      11. *-rgt-identity N/A

          \[\leadsto m \cdot \frac{\color{blue}{m}}{v} - \frac{1}{v} \cdot {m}^{3} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\frac{m}{v} \cdot m} - \frac{1}{v} \cdot {m}^{3} \]

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

   5. Simplified 99.9%

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

Alternative 7: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.6 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \mathsf{fma}\left(m, -m, m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.6e-13) (fma (/ m v) m (- m)) (* (/ m v) (fma m (- m) m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.6e-13) {
		tmp = fma((m / v), m, -m);
	} else {
		tmp = (m / v) * fma(m, -m, m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.6e-13)
		tmp = fma(Float64(m / v), m, Float64(-m));
	else
		tmp = Float64(Float64(m / v) * fma(m, Float64(-m), m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.6e-13], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * (-m) + m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.6000000000000001e-13

   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

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 91.0

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

   5. Simplified 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if 6.6000000000000001e-13 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \color{blue}{\frac{1}{m \cdot v} \cdot {m}^{3} - \frac{1}{v} \cdot {m}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{m \cdot v}} - \frac{1}{v} \cdot {m}^{3} \]

      3. cube-mult N/A

         \[\leadsto \color{blue}{\left(m \cdot \left(m \cdot m\right)\right)} \cdot \frac{1}{m \cdot v} - \frac{1}{v} \cdot {m}^{3} \]

      4. unpow2 N/A

         \[\leadsto \left(m \cdot \color{blue}{{m}^{2}}\right) \cdot \frac{1}{m \cdot v} - \frac{1}{v} \cdot {m}^{3} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{m \cdot \left({m}^{2} \cdot \frac{1}{m \cdot v}\right)} - \frac{1}{v} \cdot {m}^{3} \]

      6. associate-/r* N/A

         \[\leadsto m \cdot \left({m}^{2} \cdot \color{blue}{\frac{\frac{1}{m}}{v}}\right) - \frac{1}{v} \cdot {m}^{3} \]

      7. associate-*r/ N/A

         \[\leadsto m \cdot \color{blue}{\frac{{m}^{2} \cdot \frac{1}{m}}{v}} - \frac{1}{v} \cdot {m}^{3} \]

      8. unpow2 N/A

         \[\leadsto m \cdot \frac{\color{blue}{\left(m \cdot m\right)} \cdot \frac{1}{m}}{v} - \frac{1}{v} \cdot {m}^{3} \]

      9. associate-*l* N/A

         \[\leadsto m \cdot \frac{\color{blue}{m \cdot \left(m \cdot \frac{1}{m}\right)}}{v} - \frac{1}{v} \cdot {m}^{3} \]

      10. rgt-mult-inverse N/A

          \[\leadsto m \cdot \frac{m \cdot \color{blue}{1}}{v} - \frac{1}{v} \cdot {m}^{3} \]

      11. *-rgt-identity N/A

          \[\leadsto m \cdot \frac{\color{blue}{m}}{v} - \frac{1}{v} \cdot {m}^{3} \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\frac{m}{v} \cdot m} - \frac{1}{v} \cdot {m}^{3} \]

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

   5. Simplified 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 99.9

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.6 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \mathsf{fma}\left(m, -m, m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.6 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\frac{m}{v}, m, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m - m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.6e-13) (fma (/ m v) m (- m)) (* (/ m v) (- m (* m m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.6e-13) {
		tmp = fma((m / v), m, -m);
	} else {
		tmp = (m / v) * (m - (m * m));
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.6e-13)
		tmp = fma(Float64(m / v), m, Float64(-m));
	else
		tmp = Float64(Float64(m / v) * Float64(m - Float64(m * m)));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.6e-13], N[(N[(m / v), $MachinePrecision] * m + (-m)), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.6000000000000001e-13

   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

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 91.0

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

   5. Simplified 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if 6.6000000000000001e-13 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out-- N/A

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

      15. unsub-neg N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. +-commutative N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in m around inf

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

      1. distribute-rgt-out-- N/A

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

      2. *-lft-identity N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lft-mult-inverse N/A

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

      6. *-lft-identity N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.9

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

   8. Simplified 99.9%

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

Alternative 9: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.1

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

   5. Simplified 88.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 96.2

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

   7. Applied egg-rr 96.2%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {m}^{3}}{v}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{-1}{v} \cdot {m}^{3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v} \cdot {m}^{3} \]

      4. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot {m}^{3} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{v} \cdot {m}^{3}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{m}^{3} \cdot \frac{1}{v}}\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({m}^{3} \cdot \frac{1}{v}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{m}^{3} \cdot 1}{v}}\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{m}^{3}}}{v}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{m}^{3}}{v}}\right) \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 97.7

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

   5. Simplified 97.7%

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

Alternative 10: 87.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.1

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

   5. Simplified 88.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 96.2

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

   7. Applied egg-rr 96.2%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 0.1

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

   5. Simplified 0.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 0.1

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

   7. Applied egg-rr 0.1%

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

   9. Applied egg-rr 77.5%

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

Alternative 11: 87.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 96.2

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

   5. Simplified 96.2%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 0.1

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

   5. Simplified 0.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 0.1

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

   7. Applied egg-rr 0.1%

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

   9. Applied egg-rr 77.5%

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

4. Final simplification 87.4%

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

Alternative 12: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 13: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in m around 0

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

   1. distribute-lft-out-- N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. associate-*r* N/A

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

   10. *-inverses N/A

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

   11. associate-/l* N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. distribute-rgt-out-- N/A

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

   15. unsub-neg N/A

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

   16. mul-1-neg N/A

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

   17. +-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. +-commutative N/A

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

5. Simplified 99.8%

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

Alternative 14: 28.1% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in m around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(m\right)} \]

   2. lower-neg.f64 28.4

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

5. Simplified 28.4%

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

Reproduce

herbie shell --seed 2024207 
(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))