a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 14

Speedup: 1.0×

• 41.6% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.3e-71) (- (/ m (/ v m)) m) (/ (* m (- m (fma m m v))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.3e-71) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (m * (m - fma(m, m, v))) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.3e-71)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(Float64(m * Float64(m - fma(m, m, v))) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.3e-71], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m * N[(m - N[(m * m + v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.2999999999999999e-71

   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 80.0

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

   5. Applied rewrites 80.0%

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

   7. Applied rewrites 99.9%

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

   if 1.2999999999999999e-71 < m

   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

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 99.9%

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

Alternative 2: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+282}:\\ \;\;\;\;\frac{m \cdot \left(-m\right)}{m}\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (+ -1.0 (/ (* m (- 1.0 m)) v)))))
   (if (<= t_0 -1e+282)
     (/ (* m (- m)) m)
     (if (<= t_0 -4e-308) (- m) (* m (/ m v))))))

C

double code(double m, double v) {
	double t_0 = m * (-1.0 + ((m * (1.0 - m)) / v));
	double tmp;
	if (t_0 <= -1e+282) {
		tmp = (m * -m) / m;
	} else if (t_0 <= -4e-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 * ((-1.0d0) + ((m * (1.0d0 - m)) / v))
    if (t_0 <= (-1d+282)) then
        tmp = (m * -m) / m
    else if (t_0 <= (-4d-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 * (-1.0 + ((m * (1.0 - m)) / v));
	double tmp;
	if (t_0 <= -1e+282) {
		tmp = (m * -m) / m;
	} else if (t_0 <= -4e-308) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (-1.0 + ((m * (1.0 - m)) / v))
	tmp = 0
	if t_0 <= -1e+282:
		tmp = (m * -m) / m
	elif t_0 <= -4e-308:
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v)))
	tmp = 0.0
	if (t_0 <= -1e+282)
		tmp = Float64(Float64(m * Float64(-m)) / m);
	elseif (t_0 <= -4e-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 * (-1.0 + ((m * (1.0 - m)) / v));
	tmp = 0.0;
	if (t_0 <= -1e+282)
		tmp = (m * -m) / m;
	elseif (t_0 <= -4e-308)
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(-1.0 + N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+282], N[(N[(m * (-m)), $MachinePrecision] / m), $MachinePrecision], If[LessEqual[t$95$0, -4e-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) < -1.00000000000000003e282

   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 6.2

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

   5. Applied rewrites 6.2%

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

   7. Applied rewrites 63.9%

      \[\leadsto \frac{m \cdot \left(-m\right)}{\color{blue}{0 + m}} \]
   8. Step-by-step derivation

   9. Applied rewrites 63.9%

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

   if -1.00000000000000003e282 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -4.00000000000000013e-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 77.0

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

   5. Applied rewrites 77.0%

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

   if -4.00000000000000013e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.4%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 70.9%

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

   10. Applied rewrites 92.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -1 \cdot 10^{+282}:\\ \;\;\;\;\frac{m \cdot \left(-m\right)}{m}\\ \mathbf{elif}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ -1.0 (/ (* m (- 1.0 m)) v))) -4e-308) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -4e-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 * ((-1.0d0) + ((m * (1.0d0 - m)) / v))) <= (-4d-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 * (-1.0 + ((m * (1.0 - m)) / v))) <= -4e-308) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (-1.0 + ((m * (1.0 - m)) / v))) <= -4e-308:
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v))) <= -4e-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 * (-1.0 + ((m * (1.0 - m)) / v))) <= -4e-308)
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(-1.0 + N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-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) < -4.00000000000000013e-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 37.1

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

   5. Applied rewrites 37.1%

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

   if -4.00000000000000013e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.4%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 70.9%

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

   10. Applied rewrites 92.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \mathsf{fma}\left(-m, m, m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.8e-26) (- (/ m (/ v m)) m) (/ (* m (fma (- m) m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.8e-26) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (m * fma(-m, m, m)) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.8e-26)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(Float64(m * fma(Float64(-m), m, m)) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.8e-26], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m * N[((-m) * m + m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.8000000000000001e-26

   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 84.2

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 99.8%

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

   if 1.8000000000000001e-26 < m

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. *-lft-identity N/A

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

      11. unpow2 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

Alternative 5: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \mathsf{fma}\left(-m, m, m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.15e-26) (fma m (/ m v) (- m)) (/ (* m (fma (- m) m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.15e-26) {
		tmp = fma(m, (m / v), -m);
	} else {
		tmp = (m * fma(-m, m, m)) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.15e-26)
		tmp = fma(m, Float64(m / v), Float64(-m));
	else
		tmp = Float64(Float64(m * fma(Float64(-m), m, m)) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.15e-26], N[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision], N[(N[(m * N[((-m) * m + m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.14999999999999994e-26

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 2.14999999999999994e-26 < m

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. *-lft-identity N/A

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

      11. unpow2 N/A

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

      12. lower--.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

Alternative 6: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.15e-26) (fma m (/ m v) (- m)) (/ (* m (- m (* m m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.15e-26) {
		tmp = fma(m, (m / v), -m);
	} else {
		tmp = (m * (m - (m * m))) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.15e-26)
		tmp = fma(m, Float64(m / v), Float64(-m));
	else
		tmp = Float64(Float64(m * Float64(m - Float64(m * m))) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.15e-26], N[(m * N[(m / v), $MachinePrecision] + (-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 < 2.14999999999999994e-26

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 2.14999999999999994e-26 < m

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   6. 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. associate-/l* N/A

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

      13. unpow2 N/A

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

      14. *-lft-identity N/A

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

      15. associate-*l/ N/A

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

      19. associate-*l/ N/A

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

      20. *-lft-identity N/A

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

   7. Applied rewrites 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: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 98.3

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

   7. Applied rewrites 98.3%

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

   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. lower-/.f64 N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 98.5%

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

Alternative 8: 74.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 98.3

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

   7. Applied rewrites 98.3%

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

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.1

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

   5. Applied rewrites 6.1%

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

   7. Applied rewrites 56.3%

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

   9. Applied rewrites 56.3%

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

4. Final simplification 79.3%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. *-rgt-identity N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift--.f64 N/A

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

   17. sub-neg N/A

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

   18. +-commutative N/A

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

   19. distribute-lft-in N/A

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

   20. *-rgt-identity N/A

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

   21. lower-fma.f64 N/A

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

   22. lower-neg.f64 N/A

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

   23. lower-/.f64 N/A

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

   24. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. neg-mul-1 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 11: 74.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-neg N/A

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

      18. +-commutative N/A

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

      19. distribute-lft-in N/A

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

      20. *-rgt-identity N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-neg.f64 N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 98.3

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

   7. Applied rewrites 98.3%

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

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.1

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

   5. Applied rewrites 6.1%

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

   7. Applied rewrites 56.3%

      \[\leadsto \frac{m \cdot \left(-m\right)}{\color{blue}{0 + m}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.3%

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

Alternative 12: 74.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.6%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 98.3%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. 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 6.1

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

   5. Applied rewrites 6.1%

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

   7. Applied rewrites 56.3%

      \[\leadsto \frac{m \cdot \left(-m\right)}{\color{blue}{0 + m}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.3%

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

Alternative 13: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 14: 27.4% 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 26.6

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

5. Applied rewrites 26.6%

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

Reproduce

herbie shell --seed 2024238 
(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))