a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 8.4s

Alternatives: 14

Speedup: N/A×

• 41.4% 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, 1.1× 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 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right)\\ t_1 := m \cdot \frac{m}{v}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+79}:\\ \;\;\;\;-t\_1\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (+ -1.0 (/ (* m (- 1.0 m)) v)))) (t_1 (* m (/ m v))))
   (if (<= t_0 -5e+79) (- t_1) (if (<= t_0 -1e-306) (- m) t_1))))

C

double code(double m, double v) {
	double t_0 = m * (-1.0 + ((m * (1.0 - m)) / v));
	double t_1 = m * (m / v);
	double tmp;
	if (t_0 <= -5e+79) {
		tmp = -t_1;
	} else if (t_0 <= -1e-306) {
		tmp = -m;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = m * ((-1.0d0) + ((m * (1.0d0 - m)) / v))
    t_1 = m * (m / v)
    if (t_0 <= (-5d+79)) then
        tmp = -t_1
    else if (t_0 <= (-1d-306)) then
        tmp = -m
    else
        tmp = t_1
    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 t_1 = m * (m / v);
	double tmp;
	if (t_0 <= -5e+79) {
		tmp = -t_1;
	} else if (t_0 <= -1e-306) {
		tmp = -m;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (-1.0 + ((m * (1.0 - m)) / v))
	t_1 = m * (m / v)
	tmp = 0
	if t_0 <= -5e+79:
		tmp = -t_1
	elif t_0 <= -1e-306:
		tmp = -m
	else:
		tmp = t_1
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v)))
	t_1 = Float64(m * Float64(m / v))
	tmp = 0.0
	if (t_0 <= -5e+79)
		tmp = Float64(-t_1);
	elseif (t_0 <= -1e-306)
		tmp = Float64(-m);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (-1.0 + ((m * (1.0 - m)) / v));
	t_1 = m * (m / v);
	tmp = 0.0;
	if (t_0 <= -5e+79)
		tmp = -t_1;
	elseif (t_0 <= -1e-306)
		tmp = -m;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+79], (-t$95$1), If[LessEqual[t$95$0, -1e-306], (-m), t$95$1]]]]

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) < -5e79

   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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 0.1

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 0.1

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

   10. Applied rewrites 0.1%

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

       1. frac-2neg N/A

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

       2. neg-sub0 N/A

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

   12. Applied rewrites 77.6%

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

   if -5e79 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.00000000000000003e-306

   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 93.5

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

   5. Applied rewrites 93.5%

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

   if -1.00000000000000003e-306 < (*.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. 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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 95.3

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 89.2

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

   10. Applied rewrites 89.2%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{+79}:\\ \;\;\;\;-m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\left(-m\right) \cdot \frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ -1.0 (/ (* m (- 1.0 m)) v))) -5e+79)
   (* (- m) (/ (* m m) v))
   (fma m (/ m v) (- m))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e+79) {
		tmp = -m * ((m * m) / v);
	} else {
		tmp = fma(m, (m / v), -m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v))) <= -5e+79)
		tmp = Float64(Float64(-m) * Float64(Float64(m * m) / v));
	else
		tmp = fma(m, Float64(m / v), Float64(-m));
	end
	return 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], -5e+79], N[((-m) * N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision] + (-m)), $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) < -5e79

   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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if -5e79 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   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

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

      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. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. 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. 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 97.6

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

   7. Applied rewrites 97.6%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\left(-m\right) \cdot \frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{+79}:\\ \;\;\;\;-m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ -1.0 (/ (* m (- 1.0 m)) v))) -5e+79)
   (- (* m (/ m v)))
   (fma m (/ m v) (- m))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e+79) {
		tmp = -(m * (m / v));
	} else {
		tmp = fma(m, (m / v), -m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v))) <= -5e+79)
		tmp = Float64(-Float64(m * Float64(m / v)));
	else
		tmp = fma(m, Float64(m / v), Float64(-m));
	end
	return 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], -5e+79], (-N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), N[(m * N[(m / v), $MachinePrecision] + (-m)), $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) < -5e79

   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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 0.1

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 0.1

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

   10. Applied rewrites 0.1%

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

       1. frac-2neg N/A

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

       2. neg-sub0 N/A

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

   12. Applied rewrites 77.6%

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

   if -5e79 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   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

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

      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. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. 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. 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 97.6

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

   7. Applied rewrites 97.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{+79}:\\ \;\;\;\;-m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{+79}:\\ \;\;\;\;-m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ -1.0 (/ (* m (- 1.0 m)) v))) -5e+79)
   (- (* m (/ m v)))
   (* m (+ -1.0 (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e+79) {
		tmp = -(m * (m / v));
	} else {
		tmp = m * (-1.0 + (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((m * ((-1.0d0) + ((m * (1.0d0 - m)) / v))) <= (-5d+79)) then
        tmp = -(m * (m / v))
    else
        tmp = m * ((-1.0d0) + (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e+79) {
		tmp = -(m * (m / v));
	} else {
		tmp = m * (-1.0 + (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e+79:
		tmp = -(m * (m / v))
	else:
		tmp = m * (-1.0 + (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))) <= -5e+79)
		tmp = Float64(-Float64(m * Float64(m / v)));
	else
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -5e+79)
		tmp = -(m * (m / v));
	else
		tmp = m * (-1.0 + (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], -5e+79], (-N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $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) < -5e79

   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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 0.1

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

   7. Applied rewrites 0.1%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 0.1

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

   10. Applied rewrites 0.1%

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

       1. frac-2neg N/A

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

       2. neg-sub0 N/A

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

   12. Applied rewrites 77.6%

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

   if -5e79 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   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 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -5 \cdot 10^{+79}:\\ \;\;\;\;-m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -1 \cdot 10^{-306}:\\ \;\;\;\;m \cdot \left(-1 - \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ -1.0 (/ (* m (- 1.0 m)) v))) -1e-306)
   (* m (- -1.0 (/ m v)))
   (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -1e-306) {
		tmp = m * (-1.0 - (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 * ((-1.0d0) + ((m * (1.0d0 - m)) / v))) <= (-1d-306)) then
        tmp = m * ((-1.0d0) - (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 * (-1.0 + ((m * (1.0 - m)) / v))) <= -1e-306) {
		tmp = m * (-1.0 - (m / v));
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (-1.0 + ((m * (1.0 - m)) / v))) <= -1e-306:
		tmp = m * (-1.0 - (m / v))
	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))) <= -1e-306)
		tmp = Float64(m * Float64(-1.0 - 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 * (-1.0 + ((m * (1.0 - m)) / v))) <= -1e-306)
		tmp = m * (-1.0 - (m / v));
	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], -1e-306], N[(m * N[(-1.0 - N[(m / v), $MachinePrecision]), $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) < -1.00000000000000003e-306

   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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 36.8

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

   7. Applied rewrites 36.8%

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

      1. lift-/.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. neg-sub0 N/A

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

      5. flip3-- N/A

         \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{{0}^{3} - {m}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}}\right)\right)\right) \cdot m \]

      6. metadata-eval N/A

         \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{0} - {m}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      7. neg-sub0 N/A

         \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({m}^{3}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      8. cube-neg N/A

         \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(m\right)\right)}^{3}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      9. lift-neg.f64 N/A

         \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(\mathsf{neg}\left(m\right)\right)}}^{3}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      10. sqr-pow N/A

          \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(m\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(m\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      11. pow-prod-down N/A

          \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(m\right)\right) \cdot \left(\mathsf{neg}\left(m\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      12. lift-neg.f64 N/A

          \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(m\right)\right)} \cdot \left(\mathsf{neg}\left(m\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{{\left(\left(\mathsf{neg}\left(m\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      14. sqr-neg N/A

          \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(m \cdot m\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      15. pow-prod-down N/A

          \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{{m}^{\left(\frac{3}{2}\right)} \cdot {m}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      16. sqr-pow N/A

          \[\leadsto \left(-1 + \frac{1}{v} \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{{m}^{3}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right)\right)\right) \cdot m \]

      17. distribute-neg-frac N/A

          \[\leadsto \left(-1 + \frac{1}{v} \cdot \color{blue}{\frac{\mathsf{neg}\left({m}^{3}\right)}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}}\right) \cdot m \]

   9. Applied rewrites 83.4%

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

   if -1.00000000000000003e-306 < (*.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. 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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 95.3

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 89.2

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

   10. Applied rewrites 89.2%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -1 \cdot 10^{-306}:\\ \;\;\;\;m \cdot \left(-1 - \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ -1.0 (/ (* m (- 1.0 m)) v))) -1e-306) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m * (-1.0 + ((m * (1.0 - m)) / v))) <= -1e-306) {
		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))) <= (-1d-306)) 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))) <= -1e-306) {
		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))) <= -1e-306:
		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))) <= -1e-306)
		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))) <= -1e-306)
		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], -1e-306], (-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) < -1.00000000000000003e-306

   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 38.2

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

   5. Applied rewrites 38.2%

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

   if -1.00000000000000003e-306 < (*.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. 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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 95.3

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 89.2

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

   10. Applied rewrites 89.2%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.2e-6) (fma m (/ m v) (- m)) (/ (* (- 1.0 m) (* m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.2e-6) {
		tmp = fma(m, (m / v), -m);
	} else {
		tmp = ((1.0 - m) * (m * m)) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.2e-6)
		tmp = fma(m, Float64(m / v), Float64(-m));
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(m * m)) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.2e-6], N[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.1999999999999999e-6

   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. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. 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. 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 98.7

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

   7. Applied rewrites 98.7%

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

   if 3.1999999999999999e-6 < 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

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

      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. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.2e-6) (fma m (/ m v) (- m)) (/ (* m (- m (* m m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.2e-6) {
		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 <= 3.2e-6)
		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, 3.2e-6], 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 < 3.1999999999999999e-6

   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. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. 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. 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 98.7

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

   7. Applied rewrites 98.7%

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

   if 3.1999999999999999e-6 < 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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   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-lft-out-- N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. rgt-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r/ N/A

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

      12. *-rgt-identity N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-/l* N/A

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

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

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

      17. div-sub N/A

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

      18. unsub-neg N/A

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

      19. mul-1-neg N/A

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

   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 10: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m - m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.2e-6) (fma m (/ m v) (- m)) (* m (/ (- m (* m m)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.2e-6) {
		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 <= 3.2e-6)
		tmp = fma(m, Float64(m / v), Float64(-m));
	else
		tmp = Float64(m * Float64(Float64(m - Float64(m * m)) / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.2e-6], N[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision], N[(m * N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.1999999999999999e-6

   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. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. 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. 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 98.7

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

   7. Applied rewrites 98.7%

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

   if 3.1999999999999999e-6 < 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

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. rgt-mult-inverse N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

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

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

      16. *-lft-identity N/A

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

      17. unpow2 N/A

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

      18. lower--.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{m}{v}, -m\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m - m \cdot m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = fma(m, Float64(m / v), 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[(m * N[(m / v), $MachinePrecision] + (-m)), $MachinePrecision], (-N[(N[(m * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      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. lift--.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. neg-mul-1 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. 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. 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 97.6

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

   7. Applied rewrites 97.6%

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

   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 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 97.4%

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

Alternative 12: 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

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

   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. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. 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 13: 27.5% 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 30.4

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

5. Applied rewrites 30.4%

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

Alternative 14: 3.0% accurate, 28.0× speedup

Math

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

FPCore

(FPCore (m v) :precision binary64 m)

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return m

Julia

function code(m, v)
	return m
end

MATLAB

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

Wolfram

code[m_, v_] := m

Derivation

1. Initial program 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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 30.4

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

5. Applied rewrites 30.4%

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

   1. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - m} \]

   2. flip3-- N/A

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

   3. metadata-eval N/A

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

   4. neg-sub0 N/A

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

   5. distribute-neg-frac N/A

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

   6. sqr-pow N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{m}^{\left(\frac{3}{2}\right)} \cdot {m}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right) \]

   7. pow-prod-down N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(m \cdot m\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right) \]

   8. sqr-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(m\right)\right) \cdot \left(\mathsf{neg}\left(m\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right) \]

   9. lift-neg.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{{\left(\color{blue}{\left(\mathsf{neg}\left(m\right)\right)} \cdot \left(\mathsf{neg}\left(m\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right) \]

   10. lift-neg.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{{\left(\left(\mathsf{neg}\left(m\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(m\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right) \]

   11. pow-prod-down N/A

       \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(m\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(m\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(m \cdot m + 0 \cdot m\right)}\right) \]

   12. sqr-pow N/A

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

   13. lift-neg.f64 N/A

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

   14. cube-neg N/A

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

   15. neg-sub0 N/A

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

   16. metadata-eval N/A

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

   17. flip3-- N/A

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

   18. neg-sub0 N/A

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

   19. remove-double-neg 3.2

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

7. Applied rewrites 3.2%

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

Reproduce

herbie shell --seed 2024216 
(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))