b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

Alternatives: 17

Speedup: 1.0×

• 41.2% 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

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

FPCore

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

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (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) * (1.0d0 - m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (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) * (1.0d0 - m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- 1.0 m) (+ (/ (* m (- 1.0 m)) v) -1.0)) -0.5) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = 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 (((1.0d0 - m) * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if ((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= -0.5:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(1.0 - m) * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -0.5)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= -0.5)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(1.0 - m), $MachinePrecision] * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(m / v), $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)) (-.f64 #s(literal 1 binary64) m)) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

   5. Simplified 95.5%

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

   if -0.5 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m))

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 61.0

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

   8. Simplified 61.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;m + \mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.1e-8)
   (+ m (fma (/ m v) (fma m -2.0 1.0) -1.0))
   (/ (* (- 1.0 m) (- m (* m m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.1e-8) {
		tmp = m + fma((m / v), fma(m, -2.0, 1.0), -1.0);
	} else {
		tmp = ((1.0 - m) * (m - (m * m))) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.1e-8)
		tmp = Float64(m + fma(Float64(m / v), fma(m, -2.0, 1.0), -1.0));
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(m - Float64(m * m))) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.1e-8], N[(m + N[(N[(m / v), $MachinePrecision] * N[(m * -2.0 + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.0999999999999999e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.0999999999999999e-8 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. *-rgt-identity N/A

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

      13. unpow2 N/A

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

      14. associate--l- N/A

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

      15. lower--.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 99.0

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

   8. Simplified 99.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;m + \mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(m - m \cdot m\right) \cdot \frac{1 - m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.7e-20) (+ (/ m v) -1.0) (* (- m (* m m)) (/ (- 1.0 m) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.7e-20) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m - (m * m)) * ((1.0 - m) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.7d-20) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = (m - (m * m)) * ((1.0d0 - m) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.7e-20) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m - (m * m)) * ((1.0 - m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.7e-20:
		tmp = (m / v) + -1.0
	else:
		tmp = (m - (m * m)) * ((1.0 - m) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.7e-20)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(Float64(m - Float64(m * m)) * Float64(Float64(1.0 - m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.7e-20)
		tmp = (m / v) + -1.0;
	else
		tmp = (m - (m * m)) * ((1.0 - m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.7e-20], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6999999999999999e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 100.0

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

   8. Simplified 100.0%

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

   if 1.6999999999999999e-20 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

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

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

      12. *-rgt-identity N/A

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

      13. unpow2 N/A

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

      14. associate--l- N/A

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

      15. lower--.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. *-rgt-identity N/A

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

      9. times-frac N/A

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

      10. div-sub N/A

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

      11. unsub-neg N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. /-rgt-identity N/A

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

      15. lower-*.f64 N/A

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

   8. Simplified 99.9%

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

   9. Taylor expanded in m around inf

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

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

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lft-mult-inverse N/A

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

       5. *-lft-identity N/A

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

       6. *-lft-identity N/A

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

       7. lower--.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 99.0

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

   11. Simplified 99.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(m - m \cdot m\right) \cdot \frac{1 - m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.12 \cdot 10^{-9}:\\ \;\;\;\;m + \mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m - m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.12e-9)
   (+ m (fma (/ m v) (fma m -2.0 1.0) -1.0))
   (* (- 1.0 m) (/ (- m (* m m)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.12e-9) {
		tmp = m + fma((m / v), fma(m, -2.0, 1.0), -1.0);
	} else {
		tmp = (1.0 - m) * ((m - (m * m)) / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.12e-9)
		tmp = Float64(m + fma(Float64(m / v), fma(m, -2.0, 1.0), -1.0));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m - Float64(m * m)) / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.12e-9], N[(m + N[(N[(m / v), $MachinePrecision] * N[(m * -2.0 + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.12000000000000006e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.12000000000000006e-9 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-/l* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l/ N/A

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

      17. distribute-rgt-in N/A

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

      18. +-commutative N/A

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

      19. mul-1-neg N/A

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

      20. unsub-neg N/A

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

      21. div-sub N/A

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

   5. Simplified 99.0%

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

Alternative 6: 98.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0)
   (+ m (fma (/ m v) (fma m -2.0 1.0) -1.0))
   (* (/ (* m m) v) (+ m -2.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = m + fma((m / v), fma(m, -2.0, 1.0), -1.0);
	} else {
		tmp = ((m * m) / v) * (m + -2.0);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(m + fma(Float64(m / v), fma(m, -2.0, 1.0), -1.0));
	else
		tmp = Float64(Float64(Float64(m * m) / v) * Float64(m + -2.0));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m + N[(N[(m / v), $MachinePrecision] * N[(m * -2.0 + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision] * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 98.9

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

   5. Simplified 98.9%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

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

   5. Simplified 98.9%

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

Alternative 7: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0)
   (fma m (/ (fma m -2.0 1.0) v) -1.0)
   (* (/ (* m m) v) (+ m -2.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = fma(m, (fma(m, -2.0, 1.0) / v), -1.0);
	} else {
		tmp = ((m * m) / v) * (m + -2.0);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = fma(m, Float64(fma(m, -2.0, 1.0) / v), -1.0);
	else
		tmp = Float64(Float64(Float64(m * m) / v) * Float64(m + -2.0));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m * N[(N[(m * -2.0 + 1.0), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision] * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 98.9

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

   5. Simplified 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.8

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

   7. Applied egg-rr 98.8%

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

   8. Taylor expanded in m around 0

      \[\leadsto \mathsf{fma}\left(m, \frac{\mathsf{fma}\left(m, -2, 1\right)}{v}, \color{blue}{-1}\right) \]
   9. Step-by-step derivation

   10. Simplified 98.8%

       \[\leadsto \mathsf{fma}\left(m, \frac{\mathsf{fma}\left(m, -2, 1\right)}{v}, \color{blue}{-1}\right) \]

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

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

   5. Simplified 98.9%

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

Alternative 8: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.41:\\ \;\;\;\;\mathsf{fma}\left(m, \frac{\mathsf{fma}\left(m, -2, 1\right)}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.41) (fma m (/ (fma m -2.0 1.0) v) -1.0) (/ (* m (* m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.41) {
		tmp = fma(m, (fma(m, -2.0, 1.0) / v), -1.0);
	} else {
		tmp = (m * (m * m)) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.41)
		tmp = fma(m, Float64(fma(m, -2.0, 1.0) / v), -1.0);
	else
		tmp = Float64(Float64(m * Float64(m * m)) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.41], N[(m * N[(N[(m * -2.0 + 1.0), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(m * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.409999999999999976

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 98.9

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

   5. Simplified 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.8

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

   7. Applied egg-rr 98.8%

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

   8. Taylor expanded in m around 0

      \[\leadsto \mathsf{fma}\left(m, \frac{\mathsf{fma}\left(m, -2, 1\right)}{v}, \color{blue}{-1}\right) \]
   9. Step-by-step derivation

   10. Simplified 98.8%

       \[\leadsto \mathsf{fma}\left(m, \frac{\mathsf{fma}\left(m, -2, 1\right)}{v}, \color{blue}{-1}\right) \]

   if 0.409999999999999976 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

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

      1. lower-/.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.2

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

   5. Simplified 98.2%

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

Alternative 9: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $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 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 97.5

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

   5. Simplified 97.5%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

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

      1. lower-/.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 97.8%

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

Alternative 10: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

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

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

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

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

   12. *-rgt-identity N/A

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

   13. unpow2 N/A

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

   14. associate--l- N/A

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

   15. lower--.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-fma.f64 99.9

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

5. Simplified 99.9%

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

Alternative 11: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6) (+ (/ m v) -1.0) (/ (* m (* m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m * (m * m)) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.6d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = (m * (m * m)) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m * (m * m)) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6:
		tmp = (m / v) + -1.0
	else:
		tmp = (m * (m * m)) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(Float64(m * Float64(m * m)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6)
		tmp = (m / v) + -1.0;
	else
		tmp = (m * (m * m)) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(m * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000009

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 96.8

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

   8. Simplified 96.8%

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

   if 2.60000000000000009 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

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

      1. lower-/.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.8

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

   5. Simplified 98.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6) (+ (/ m v) -1.0) (* m (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.6d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = m * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6:
		tmp = (m / v) + -1.0
	else:
		tmp = m * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(m * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6)
		tmp = (m / v) + -1.0;
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000009

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in v around 0

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

      1. lower-/.f64 96.8

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

   8. Simplified 96.8%

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

   if 2.60000000000000009 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

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

      1. lower-/.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.8

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

   5. Simplified 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.8

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

   7. Applied egg-rr 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 98.8

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

   9. Applied egg-rr 98.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, m, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.35e+154) (+ -1.0 (+ m (/ m v))) (fma m m -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.35e+154) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = fma(m, m, -1.0);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.35e+154)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = fma(m, m, -1.0);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.35e+154], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * m + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.35000000000000003e154

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 72.6

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

   5. Simplified 72.6%

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

   if 1.35000000000000003e154 < m

   1. Initial program 100.0%

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

   3. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(1 - m\right)} \]

      3. associate--r- N/A

         \[\leadsto \color{blue}{\left(0 - 1\right) + m} \]

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 7.2

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

   5. Simplified 7.2%

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

      1. flip-+ N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in m around 0

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

   10. Simplified 100.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, m, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. lower-+.f64 N/A

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

   10. lower-/.f64 73.0

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

5. Simplified 73.0%

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

Alternative 15: 76.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{m}{v} + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. lower-+.f64 N/A

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

   10. lower-/.f64 73.0

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

5. Simplified 73.0%

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

6. Taylor expanded in v around 0

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

   1. lower-/.f64 73.0

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

8. Simplified 73.0%

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

9. Final simplification 73.0%

   \[\leadsto \frac{m}{v} + -1 \]
10. Add Preprocessing

Alternative 16: 27.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ m + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in v around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \left(1 - m\right)} \]

   3. associate--r- N/A

      \[\leadsto \color{blue}{\left(0 - 1\right) + m} \]

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 28.2

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

5. Simplified 28.2%

   \[\leadsto \color{blue}{m + -1} \]
6. Add Preprocessing

Alternative 17: 24.9% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

5. Simplified 25.6%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (m v)
  :name "b 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) (- 1.0 m)))