b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.0s

Alternatives: 11

Speedup: 1.0×

• 41.9% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 73.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(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)) (-.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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

   7. Applied rewrites 95.9%

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

   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

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. unsub-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft1-in N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{0 - \left(1 - m\right)}\right) \]

      19. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{\left(0 - 1\right) + m}\right) \]

      20. metadata-eval N/A

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

      21. +-commutative N/A

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

      22. metadata-eval N/A

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

      23. sub-neg N/A

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

      24. lower--.f64 37.1

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

   5. Applied rewrites 37.1%

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

   6. Taylor expanded in m around inf

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

   8. Applied rewrites 34.9%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 68.8%

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

4. Final simplification 74.9%

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

Alternative 3: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - m\right) \cdot m\right) \cdot \frac{1 - m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.6e-8)
   (fma (fma -2.0 m 1.0) (/ m v) (- m 1.0))
   (* (* (- 1.0 m) m) (/ (- 1.0 m) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.6e-8) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), (m - 1.0));
	} else {
		tmp = ((1.0 - m) * m) * ((1.0 - m) / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.6e-8)
		tmp = fma(fma(-2.0, m, 1.0), Float64(m / v), Float64(m - 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 - m) * m) * Float64(Float64(1.0 - m) / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.6e-8], N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + N[(m - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.6000000000000002e-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-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. unsub-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft1-in N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{0 - \left(1 - m\right)}\right) \]

      19. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{\left(0 - 1\right) + m}\right) \]

      20. metadata-eval N/A

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

      21. +-commutative N/A

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

      22. metadata-eval N/A

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

      23. sub-neg N/A

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

      24. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 4.6000000000000002e-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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in m around inf

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

   12. Applied rewrites 99.7%

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

Alternative 4: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-m\right) \cdot m}{v} \cdot \left(1 - m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.62)
   (fma (fma -2.0 m 1.0) (/ m v) (- m 1.0))
   (* (/ (* (- m) m) v) (- 1.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.62) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), (m - 1.0));
	} else {
		tmp = ((-m * m) / v) * (1.0 - m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.62)
		tmp = fma(fma(-2.0, m, 1.0), Float64(m / v), Float64(m - 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(-m) * m) / v) * Float64(1.0 - m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.62], N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + N[(m - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-m) * m), $MachinePrecision] / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.619999999999999996

   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

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. unsub-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft1-in N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{0 - \left(1 - m\right)}\right) \]

      19. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{\left(0 - 1\right) + m}\right) \]

      20. metadata-eval N/A

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

      21. +-commutative N/A

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

      22. metadata-eval N/A

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

      23. sub-neg N/A

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

      24. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if 0.619999999999999996 < 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 m around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 5: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.62:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-m\right) \cdot m\right) \cdot \frac{1 - m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.62)
   (fma (fma -2.0 m 1.0) (/ m v) (- m 1.0))
   (* (* (- m) m) (/ (- 1.0 m) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.62) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), (m - 1.0));
	} else {
		tmp = (-m * m) * ((1.0 - m) / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.62)
		tmp = fma(fma(-2.0, m, 1.0), Float64(m / v), Float64(m - 1.0));
	else
		tmp = Float64(Float64(Float64(-m) * m) * Float64(Float64(1.0 - m) / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.62], N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + N[(m - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-m) * m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.619999999999999996

   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

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l- N/A

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

      6. unsub-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft1-in N/A

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{0 - \left(1 - m\right)}\right) \]

      19. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, \color{blue}{\left(0 - 1\right) + m}\right) \]

      20. metadata-eval N/A

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

      21. +-commutative N/A

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

      22. metadata-eval N/A

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

      23. sub-neg N/A

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

      24. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if 0.619999999999999996 < 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in m around inf

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

   12. Applied rewrites 99.0%

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

Alternative 6: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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

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

      1. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if 1 < 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in m around inf

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

   12. Applied rewrites 99.0%

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

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Final simplification 99.9%

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

Alternative 8: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 9: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.4e+154) (- (+ (/ m v) m) 1.0) (/ (* m m) 1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.4e+154) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = (m * m) / 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.4d+154) then
        tmp = ((m / v) + m) - 1.0d0
    else
        tmp = (m * m) / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.4e+154) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = (m * m) / 1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.4e+154:
		tmp = ((m / v) + m) - 1.0
	else:
		tmp = (m * m) / 1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.4e+154)
		tmp = Float64(Float64(Float64(m / v) + m) - 1.0);
	else
		tmp = Float64(Float64(m * m) / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.4e+154)
		tmp = ((m / v) + m) - 1.0;
	else
		tmp = (m * m) / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.4e+154], N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(m * m), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.4e154

   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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if 1.4e154 < 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. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 6.6

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

   5. Applied rewrites 6.6%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in m around inf

       \[\leadsto \frac{{m}^{2}}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 100.0%

       \[\leadsto \frac{m \cdot m}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 26.6% 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. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 24.4

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

5. Applied rewrites 24.4%

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

Alternative 11: 24.1% 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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in m around 0

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

7. Applied rewrites 22.1%

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

Reproduce

herbie shell --seed 2024331 
(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)))