b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.8%

Time: 4.6s

Alternatives: 11

Speedup: 1.0×

• 42.5% 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: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(\left(1 - m\right) \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.9e-15)
   (* (- (/ m v) 1.0) (- 1.0 m))
   (/ (* (- 1.0 m) (* (- 1.0 m) m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-15) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((1.0 - m) * ((1.0 - m) * m)) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.9d-15) then
        tmp = ((m / v) - 1.0d0) * (1.0d0 - m)
    else
        tmp = ((1.0d0 - m) * ((1.0d0 - m) * m)) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-15) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((1.0 - m) * ((1.0 - m) * m)) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.9e-15:
		tmp = ((m / v) - 1.0) * (1.0 - m)
	else:
		tmp = ((1.0 - m) * ((1.0 - m) * m)) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.9e-15)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * Float64(1.0 - m));
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(Float64(1.0 - m) * m)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.9e-15)
		tmp = ((m / v) - 1.0) * (1.0 - m);
	else
		tmp = ((1.0 - m) * ((1.0 - m) * m)) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.9e-15], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.90000000000000019e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 2.90000000000000019e-15 < 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

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in v around 0

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

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 99.9%

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

Alternative 2: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(1 + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right)\right) - 1} \]
   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. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      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 98.7

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

   5. Applied rewrites 98.7%

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

   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 0

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. distribute-lft-in N/A

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

      6. distribute-lft-neg-in N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. times-frac N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. rgt-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

      17. *-lft-identity N/A

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

      18. mul-1-neg N/A

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

   8. Applied rewrites 98.1%

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

   10. Applied rewrites 98.2%

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

Alternative 3: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(1 + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right)\right) - 1} \]
   7. 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. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 98.5

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

   8. Applied rewrites 98.5%

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

   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 0

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. distribute-lft-in N/A

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

      6. distribute-lft-neg-in N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. times-frac N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. rgt-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

      17. *-lft-identity N/A

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

      18. mul-1-neg N/A

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

   8. Applied rewrites 98.1%

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

   10. Applied rewrites 98.2%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-2 + m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- (/ m v) 1.0) (- 1.0 m)) (/ (* (+ -2.0 m) (* m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((-2.0 + 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 <= 1.6d0) then
        tmp = ((m / v) - 1.0d0) * (1.0d0 - m)
    else
        tmp = (((-2.0d0) + m) * (m * m)) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((-2.0 + m) * (m * m)) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = ((m / v) - 1.0) * (1.0 - m)
	else:
		tmp = ((-2.0 + m) * (m * m)) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * Float64(1.0 - m));
	else
		tmp = Float64(Float64(Float64(-2.0 + m) * Float64(m * m)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = ((m / v) - 1.0) * (1.0 - m);
	else
		tmp = ((-2.0 + m) * (m * m)) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 + m), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 1.6000000000000001 < 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

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. distribute-lft-in N/A

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

      6. distribute-lft-neg-in N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. times-frac N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. rgt-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

      17. *-lft-identity N/A

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

      18. mul-1-neg N/A

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

   8. Applied rewrites 98.1%

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

   10. Applied rewrites 98.2%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{m + -2}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- (/ m v) 1.0) (- 1.0 m)) (* (* m m) (/ (+ m -2.0) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = (m * m) * ((m + -2.0) / 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 <= 1.6d0) then
        tmp = ((m / v) - 1.0d0) * (1.0d0 - m)
    else
        tmp = (m * m) * ((m + (-2.0d0)) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = (m * m) * ((m + -2.0) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = ((m / v) - 1.0) * (1.0 - m)
	else:
		tmp = (m * m) * ((m + -2.0) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * Float64(1.0 - m));
	else
		tmp = Float64(Float64(m * m) * Float64(Float64(m + -2.0) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = ((m / v) - 1.0) * (1.0 - m);
	else
		tmp = (m * m) * ((m + -2.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], N[(N[(m * m), $MachinePrecision] * N[(N[(m + -2.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 1.6000000000000001 < 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

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]
   7. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. distribute-lft-in N/A

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

      6. distribute-lft-neg-in N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. times-frac N/A

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

      12. *-rgt-identity N/A

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

      13. associate-*r/ N/A

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

      14. rgt-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

      17. *-lft-identity N/A

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

      18. mul-1-neg N/A

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

   8. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(m \cdot m\right) \cdot \frac{m + -2}{v}} \]
3. Recombined 2 regimes into one program.
4. 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: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 22.3%

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

6. Taylor expanded in v around 0

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

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in m around 0

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

11. Applied rewrites 99.9%

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

Alternative 8: 87.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. 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 48.1

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

5. Applied rewrites 48.1%

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

   1. lift--.f64 N/A

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

   2. *-lft-identity N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. distribute-lft-neg-out N/A

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

   10. mul-1-neg N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. *-lft-identity N/A

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

   16. unpow1 N/A

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

   17. metadata-eval N/A

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

   18. sqrt-pow1 N/A

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

   19. pow2 N/A

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

   20. sqr-neg-rev N/A

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

   21. mul-1-neg N/A

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

   22. mul-1-neg N/A

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

   23. pow2 N/A

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

   24. sqrt-pow1 N/A

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

   25. metadata-eval N/A

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

   26. metadata-eval N/A

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

   27. unpow-prod-down N/A

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

7. Applied rewrites 84.9%

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

Alternative 9: 75.2% 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 72.5

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

5. Applied rewrites 72.5%

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

Alternative 10: 26.7% 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 24.7

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

5. Applied rewrites 24.7%

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

Alternative 11: 24.3% 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 22.3%

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

Reproduce

herbie shell --seed 2024332 
(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)))