b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 10

Speedup: N/A×

• 42.3% 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, 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 2: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 98.3%

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

   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. 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 inf 98.3%

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

      1. div-inv 98.2%

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{v}} \]

      2. cube-mult 98.2%

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

      3. associate-*l* 98.2%

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

      4. add-sqr-sqrt 98.2%

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

      5. sqrt-unprod 98.2%

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

      6. sqr-neg 98.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.1%

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

      9. div-inv 0.1%

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

      10. associate-*l/ 0.1%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 98.2%

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

      13. sqr-neg 98.2%

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

      14. sqrt-unprod 98.2%

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

      15. add-sqr-sqrt 98.2%

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

      16. associate-*l/ 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. associate-/l* 98.2%

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

      2. associate-/r/ 98.2%

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

   8. Applied egg-rr 98.2%

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

4. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = (m / v) * (m * (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * N[(m + -2.0), $MachinePrecision]), $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. Taylor expanded in m around 0 98.3%

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

   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. 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. Step-by-step derivation

      1. associate-/r/ 99.9%

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

      2. div-inv 99.9%

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

      3. associate-/r* 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 23.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   7. Step-by-step derivation

      1. unpow2 23.4%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. cube-mult 23.4%

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

      3. associate-*r/ 23.3%

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

      4. distribute-rgt-out 99.0%

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

      5. associate-/l* 99.0%

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

   8. Simplified 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*r/ 99.0%

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

      3. +-commutative 99.0%

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

   10. Applied egg-rr 99.0%

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

       1. clear-num 99.0%

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

       2. associate-/r/ 99.0%

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

       3. clear-num 99.0%

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

       4. *-commutative 99.0%

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

   12. Applied egg-rr 99.0%

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

4. Final simplification 98.6%

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

Alternative 4: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m + -2.0), $MachinePrecision] * N[(m / N[(v / m), $MachinePrecision]), $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. Taylor expanded in m around 0 98.3%

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

   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. 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. Step-by-step derivation

      1. associate-/r/ 99.9%

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

      2. div-inv 99.9%

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

      3. associate-/r* 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 23.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   7. Step-by-step derivation

      1. unpow2 23.4%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. cube-mult 23.4%

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

      3. associate-*r/ 23.3%

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

      4. distribute-rgt-out 99.0%

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

      5. associate-/l* 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 98.6%

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

Alternative 5: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision] / N[(v / m), $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. Taylor expanded in m around 0 98.3%

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

   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. 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. Step-by-step derivation

      1. associate-/r/ 99.9%

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

      2. div-inv 99.9%

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

      3. associate-/r* 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 23.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   7. Step-by-step derivation

      1. unpow2 23.4%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. cube-mult 23.4%

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

      3. associate-*r/ 23.3%

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

      4. distribute-rgt-out 99.0%

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

      5. associate-/l* 99.0%

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

   8. Simplified 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*r/ 99.0%

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

      3. +-commutative 99.0%

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

   10. Applied egg-rr 99.0%

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

4. Final simplification 98.6%

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

Alternative 6: 62.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.65 \cdot 10^{-187}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 4.1 \cdot 10^{-132}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.65e-187)
   -1.0
   (if (<= m 5.2e-165) (/ m v) (if (<= m 4.1e-132) -1.0 (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.65e-187) {
		tmp = -1.0;
	} else if (m <= 5.2e-165) {
		tmp = m / v;
	} else if (m <= 4.1e-132) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.65d-187) then
        tmp = -1.0d0
    else if (m <= 5.2d-165) then
        tmp = m / v
    else if (m <= 4.1d-132) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.65e-187) {
		tmp = -1.0;
	} else if (m <= 5.2e-165) {
		tmp = m / v;
	} else if (m <= 4.1e-132) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.65e-187:
		tmp = -1.0
	elif m <= 5.2e-165:
		tmp = m / v
	elif m <= 4.1e-132:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.65e-187)
		tmp = -1.0;
	elseif (m <= 5.2e-165)
		tmp = Float64(m / v);
	elseif (m <= 4.1e-132)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.65e-187)
		tmp = -1.0;
	elseif (m <= 5.2e-165)
		tmp = m / v;
	elseif (m <= 4.1e-132)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.65e-187], -1.0, If[LessEqual[m, 5.2e-165], N[(m / v), $MachinePrecision], If[LessEqual[m, 4.1e-132], -1.0, N[(m / v), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if m < 1.65e-187 or 5.20000000000000015e-165 < m < 4.10000000000000007e-132

   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 76.6%

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

   if 1.65e-187 < m < 5.20000000000000015e-165 or 4.10000000000000007e-132 < 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 37.2%

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

   3. Taylor expanded in m around inf 26.8%

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

      1. +-commutative 26.8%

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

      2. mul-1-neg 26.8%

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

      3. unpow2 26.8%

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

      4. unsub-neg 26.8%

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

      5. *-commutative 26.8%

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

      6. distribute-lft-in 26.8%

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

      7. *-rgt-identity 26.8%

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

      8. associate-*r/ 26.9%

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

      9. *-rgt-identity 26.9%

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

      10. associate-/l* 26.9%

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

   5. Simplified 26.9%

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

   6. Taylor expanded in v around 0 26.9%

      \[\leadsto \color{blue}{\frac{m - {m}^{2}}{v}} \]
   7. Step-by-step derivation

      1. unpow2 26.9%

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

   8. Simplified 26.9%

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

   9. Taylor expanded in m around 0 61.7%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.65 \cdot 10^{-187}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 5.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 4.1 \cdot 10^{-132}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 7: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.38) (+ (/ m v) -1.0) (* m (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 0.38d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = m * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.38:
		tmp = (m / v) + -1.0
	else:
		tmp = m * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.38)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(m * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.38)
		tmp = (m / v) + -1.0;
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.38], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.38

   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. Step-by-step derivation

      1. associate-/r/ 100.0%

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

      2. div-inv 100.0%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in m around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. *-commutative 98.0%

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

      3. distribute-lft-in 98.0%

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

      4. *-rgt-identity 98.0%

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

      5. associate-*r/ 98.3%

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

      6. *-rgt-identity 98.3%

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

      7. metadata-eval 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in v around 0 98.3%

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

   if 0.38 < m

   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 inf 98.3%

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

      1. div-inv 98.2%

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{v}} \]

      2. cube-mult 98.2%

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

      3. associate-*l* 98.2%

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

      4. add-sqr-sqrt 98.2%

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

      5. sqrt-unprod 98.2%

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

      6. sqr-neg 98.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.1%

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

      9. div-inv 0.1%

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

      10. associate-*l/ 0.1%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 98.2%

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

      13. sqr-neg 98.2%

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

      14. sqrt-unprod 98.2%

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

      15. add-sqr-sqrt 98.2%

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

      16. associate-*l/ 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. associate-/l* 98.2%

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

      2. associate-/r/ 98.2%

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

   8. Applied egg-rr 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 8: 75.6% 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. Step-by-step derivation

   1. associate-/r/ 99.9%

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

   2. div-inv 99.9%

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

   3. associate-/r* 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in m around 0 78.7%

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

   1. sub-neg 78.7%

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

   2. *-commutative 78.7%

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

   3. distribute-lft-in 78.7%

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

   4. *-rgt-identity 78.7%

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

   5. associate-*r/ 78.8%

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

   6. *-rgt-identity 78.8%

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

   7. metadata-eval 78.8%

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

8. Simplified 78.8%

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

9. Taylor expanded in v around 0 78.8%

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

10. Final simplification 78.8%

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

Alternative 9: 27.0% 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 28.1%

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

   1. neg-mul-1 28.1%

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

   2. neg-sub0 28.1%

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

   3. associate--r- 28.1%

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

   4. metadata-eval 28.1%

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

6. Simplified 28.1%

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

7. Final simplification 28.1%

   \[\leadsto m + -1 \]

Alternative 10: 24.5% 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 25.8%

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

5. Final simplification 25.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023207 
(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)))