b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 4.5s

Alternatives: 13

Speedup: 1.0×

• 41.3% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot m}{v} \cdot \left(1 - m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.25e-11)
   (- (fma (fma -2.0 m 1.0) (/ m v) m) 1.0)
   (* (/ (* (- 1.0 m) m) v) (- 1.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-11) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), m) - 1.0;
	} else {
		tmp = (((1.0 - m) * m) / v) * (1.0 - m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.25e-11)
		tmp = Float64(fma(fma(-2.0, m, 1.0), Float64(m / v), m) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) * m) / v) * Float64(1.0 - m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.25e-11], N[(N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.25000000000000005e-11

   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. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1.25000000000000005e-11 < 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

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, m, 1\right), \frac{m}{v}, m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} \cdot \left(1 - m\right)\right) \cdot \left(1 - m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.95e-8)
   (- (fma (fma -2.0 m 1.0) (/ m v) m) 1.0)
   (* (* (/ m v) (- 1.0 m)) (- 1.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.95e-8) {
		tmp = fma(fma(-2.0, m, 1.0), (m / v), m) - 1.0;
	} else {
		tmp = ((m / v) * (1.0 - m)) * (1.0 - m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.95e-8)
		tmp = Float64(fma(fma(-2.0, m, 1.0), Float64(m / v), m) - 1.0);
	else
		tmp = Float64(Float64(Float64(m / v) * Float64(1.0 - m)) * Float64(1.0 - m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.95e-8], N[(N[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.94999999999999992e-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. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1.94999999999999992e-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

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

Alternative 3: 98.2% accurate, 0.9× speedup

Math

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

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 = Float64(fma(fma(-2.0, m, 1.0), Float64(m / v), 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[(N[(-2.0 * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $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-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. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.619999999999999996 < 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

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{{m}^{2}}{v}\right)} \cdot \left(1 - m\right) \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 4: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.409999999999999976

   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-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. +-commutative N/A

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

      4. +-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 99.0

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

   5. Applied rewrites 99.0%

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

   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. lower-pow.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 99.1%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-inverses N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 97.0%

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

   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. lower-pow.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 97.9%

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

Alternative 6: 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]

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. Add Preprocessing

Alternative 7: 97.6% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0) (* (- (/ m v) 1.0) (- 1.0 m)) (* m (/ (* m 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 * 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 * 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 * 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 * 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(m * Float64(Float64(m * 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 * 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[(m * N[(N[(m * 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 97.0

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

   5. Applied rewrites 97.0%

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

   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. lower-pow.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 97.9%

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

Alternative 8: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

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 / 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 (m <= 1.0d0) then
        tmp = ((m / v) - 1.0d0) * (1.0d0 - m)
    else
        tmp = (m * (m / v)) * m
    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 / v)) * m;
	}
	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 / v)) * m
	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(m * Float64(m / v)) * m);
	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 / v)) * m;
	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 * N[(m / v), $MachinePrecision]), $MachinePrecision] * m), $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 97.0

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

   5. Applied rewrites 97.0%

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

   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. lower-pow.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

Alternative 9: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[m_, v_] := N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m / 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. Taylor expanded in m around 0

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

   1. +-commutative N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. *-inverses N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-/.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 10: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.5:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.5) (- (+ (/ m v) m) 1.0) (* (* m (/ m v)) m)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.5) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = (m * (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 (m <= 2.5d0) then
        tmp = ((m / v) + m) - 1.0d0
    else
        tmp = (m * (m / v)) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.5) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = (m * (m / v)) * m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.5:
		tmp = ((m / v) + m) - 1.0
	else:
		tmp = (m * (m / v)) * m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.5)
		tmp = Float64(Float64(Float64(m / v) + m) - 1.0);
	else
		tmp = Float64(Float64(m * Float64(m / v)) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.5)
		tmp = ((m / v) + m) - 1.0;
	else
		tmp = (m * (m / v)) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.5], N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 2.5 < 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. lower-pow.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

Alternative 11: 75.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(N[(m / v), $MachinePrecision] + m), $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. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. lower--.f64 N/A

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 76.8

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

5. Applied rewrites 76.8%

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

Alternative 12: 26.6% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(-1.0 + m), $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. *-lft-identity N/A

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

   3. metadata-eval N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. distribute-neg-in N/A

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

   6. metadata-eval N/A

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

   7. mul-1-neg N/A

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

   8. remove-double-neg N/A

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

   9. lower-+.f64 25.8

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

5. Applied rewrites 25.8%

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

Alternative 13: 24.2% 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. Applied rewrites 23.8%

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

Reproduce

herbie shell --seed 2024326 
(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)))