b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 5.2s

Alternatives: 9

Speedup: 1.0×

• 41.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (+ (/ m v) -1.0) (/ m (/ v (+ m 1.0)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m / (v / (m + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.4d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = m / (v / (m + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m / (v / (m + 1.0));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = (m / v) + -1.0
	else:
		tmp = m / (v / (m + 1.0))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(m / Float64(v / Float64(m + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = (m / v) + -1.0;
	else
		tmp = m / (v / (m + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(m / N[(v / N[(m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. distribute-lft-in 97.9%

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

      3. un-div-inv 98.1%

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

      4. *-rgt-identity 98.1%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in v around 0 98.1%

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

   if 2.39999999999999991 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-lft-in 0.1%

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

      5. *-commutative 0.1%

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

      6. associate-*l/ 0.1%

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

      7. *-commutative 0.1%

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

      8. *-un-lft-identity 0.1%

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

      9. *-un-lft-identity 0.1%

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

      10. sub-neg 0.1%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 75.5%

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

      13. sqr-neg 75.5%

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

      14. sqrt-unprod 75.5%

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

      15. add-sqr-sqrt 75.5%

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

      16. +-commutative 75.5%

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

      17. distribute-lft1-in 75.5%

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

      18. fma-def 75.5%

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

   4. Applied egg-rr 75.5%

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

      1. *-lft-identity 75.5%

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

      2. *-lft-identity 75.5%

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

      3. fma-udef 75.5%

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

      4. distribute-lft1-in 75.5%

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

      5. +-commutative 75.5%

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

      6. associate-*r/ 75.5%

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

      7. distribute-lft-in 75.5%

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

      8. +-commutative 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in v around 0 75.5%

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

      1. associate-/l* 75.5%

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

      2. +-commutative 75.5%

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

   9. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{m}{\frac{v}{m + 1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m + 1}}\\ \end{array} \]

Alternative 4: 87.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in m around 0 45.7%

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

   1. *-commutative 45.7%

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

   2. sub-neg 45.7%

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

   3. metadata-eval 45.7%

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

   4. distribute-lft-in 45.7%

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

   5. *-commutative 45.7%

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

   6. associate-*l/ 45.7%

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

   7. *-commutative 45.7%

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

   8. *-un-lft-identity 45.7%

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

   9. *-un-lft-identity 45.7%

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

   10. sub-neg 45.7%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 86.0%

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

   13. sqr-neg 86.0%

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

   14. sqrt-unprod 86.0%

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

   15. add-sqr-sqrt 86.0%

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

   16. +-commutative 86.0%

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

   17. distribute-lft1-in 86.0%

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

   18. fma-def 86.0%

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

4. Applied egg-rr 86.0%

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

   1. *-lft-identity 86.0%

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

   2. *-lft-identity 86.0%

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

   3. fma-udef 86.0%

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

   4. distribute-lft1-in 86.0%

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

   5. +-commutative 86.0%

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

   6. associate-*r/ 86.0%

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

   7. distribute-lft-in 86.0%

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

   8. +-commutative 86.0%

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

6. Simplified 86.0%

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

7. Final simplification 86.0%

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

Alternative 5: 61.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.36 \cdot 10^{-192}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.36e-192) -1.0 (+ m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.36e-192) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.36d-192) then
        tmp = -1.0d0
    else
        tmp = m + (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.36e-192) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.36e-192:
		tmp = -1.0
	else:
		tmp = m + (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.36e-192)
		tmp = -1.0;
	else
		tmp = Float64(m + Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.36e-192)
		tmp = -1.0;
	else
		tmp = m + (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.36e-192], -1.0, N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.3600000000000001e-192

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 85.6%

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

   if 1.3600000000000001e-192 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 33.1%

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

   3. Taylor expanded in m around inf 22.1%

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

      1. neg-mul-1 22.1%

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

      2. +-commutative 22.1%

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

      3. distribute-rgt-in 22.1%

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

      4. *-lft-identity 22.1%

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

      5. associate-+l+ 22.1%

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

      6. associate-*l/ 22.2%

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

      7. *-lft-identity 22.2%

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

      8. sub-neg 22.2%

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

      9. div-sub 22.2%

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

      10. unpow2 22.2%

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

      11. cancel-sign-sub-inv 22.2%

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

      12. *-lft-identity 22.2%

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

      13. mul-1-neg 22.2%

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

      14. distribute-rgt-in 22.2%

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

      15. mul-1-neg 22.2%

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

      16. sub-neg 22.2%

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

      17. associate-/l* 22.2%

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

   5. Simplified 22.2%

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

   6. Taylor expanded in m around 0 59.0%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.36 \cdot 10^{-192}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \]

Alternative 6: 75.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 75.5%

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

   1. +-commutative 75.5%

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

   2. distribute-lft-in 75.5%

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

   3. un-div-inv 75.6%

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

   4. *-rgt-identity 75.6%

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

6. Applied egg-rr 75.6%

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

7. Final simplification 75.6%

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

Alternative 7: 75.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 75.5%

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

   1. +-commutative 75.5%

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

   2. distribute-lft-in 75.5%

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

   3. un-div-inv 75.6%

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

   4. *-rgt-identity 75.6%

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

6. Applied egg-rr 75.6%

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

7. Taylor expanded in v around 0 75.6%

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

8. Final simplification 75.6%

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

Alternative 8: 26.6% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around inf 26.8%

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

   1. neg-mul-1 26.8%

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

   2. neg-sub0 26.8%

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

   3. associate--r- 26.8%

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

   4. metadata-eval 26.8%

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

6. Simplified 26.8%

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

7. Final simplification 26.8%

   \[\leadsto m + -1 \]

Alternative 9: 24.2% accurate, 13.0× 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. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 24.1%

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

5. Final simplification 24.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023318 
(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)))