b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 11

Speedup: N/A×

• 40.6% 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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. associate-*r/ 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 87.4% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0)
   (* (- 1.0 m) (+ (/ m v) -1.0))
   (+ (+ m -1.0) (* m (/ (+ m -1.0) 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 + -1.0) + (m * ((m + -1.0) / 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 <= 1.0d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = (m + (-1.0d0)) + (m * ((m + (-1.0d0)) / 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 + -1.0) + (m * ((m + -1.0) / 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 + -1.0) + (m * ((m + -1.0) / 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(Float64(m + -1.0) + Float64(m * Float64(Float64(m + -1.0) / 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 + -1.0) + (m * ((m + -1.0) / 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[(N[(m + -1.0), $MachinePrecision] + N[(m * N[(N[(m + -1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 98.1%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]
   4. 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-rgt-in 0.1%

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

      5. associate-*l/ 0.1%

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

      6. associate-/l* 0.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 76.5%

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

      9. sqr-neg 76.5%

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

      10. sqrt-unprod 76.5%

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

      11. add-sqr-sqrt 76.5%

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

      12. neg-sub0 76.5%

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

      13. associate--r- 76.5%

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

      14. metadata-eval 76.5%

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

      15. neg-mul-1 76.5%

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

      16. neg-sub0 76.5%

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

      17. associate--r- 76.5%

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

      18. metadata-eval 76.5%

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

   5. Applied egg-rr 76.5%

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

4. Final simplification 88.3%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4 \cdot 10^{-39}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 + m \cdot \left(m - 2\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4e-39) (+ (/ m v) -1.0) (/ (* m (+ 1.0 (* m (- m 2.0)))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4e-39) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m * (1.0 + (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 <= 4d-39) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = (m * (1.0d0 + (m * (m - 2.0d0)))) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4e-39) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m * (1.0 + (m * (m - 2.0)))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4e-39:
		tmp = (m / v) + -1.0
	else:
		tmp = (m * (1.0 + (m * (m - 2.0)))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4e-39)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(Float64(m * Float64(1.0 + Float64(m * Float64(m - 2.0)))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4e-39)
		tmp = (m / v) + -1.0;
	else
		tmp = (m * (1.0 + (m * (m - 2.0)))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4e-39], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(m * N[(1.0 + N[(m * N[(m - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.99999999999999972e-39

   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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.0%

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

   6. Taylor expanded in v around 0 100.0%

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

   7. Taylor expanded in m around 0 100.0%

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

   if 3.99999999999999972e-39 < 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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.8%

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

   6. Taylor expanded in v around 0 99.9%

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

   7. Taylor expanded in v around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4 \cdot 10^{-39}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 + m \cdot \left(m - 2\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0) (* (- 1.0 m) (+ (/ m v) -1.0)) (* m (/ (+ m -1.0) 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 + -1.0) / 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 <= 1.0d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = m * ((m + (-1.0d0)) / 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 + -1.0) / 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 + -1.0) / 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(Float64(m + -1.0) / 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 + -1.0) / 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[(N[(m + -1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 98.1%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

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

      1. pow1 0.1%

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

      2. *-commutative 0.1%

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

      3. sub-neg 0.1%

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

      4. metadata-eval 0.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 76.5%

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

      7. sqr-neg 76.5%

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

      8. sqrt-unprod 76.5%

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

      9. add-sqr-sqrt 76.5%

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

      10. neg-sub0 76.5%

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

      11. associate--r- 76.5%

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

      12. metadata-eval 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. unpow1 76.5%

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

      2. +-commutative 76.5%

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

   7. Simplified 76.5%

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

   8. Taylor expanded in v around 0 76.5%

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

      1. sub-neg 76.5%

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

      2. metadata-eval 76.5%

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

      3. associate-*r/ 76.5%

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

   10. Simplified 76.5%

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

4. Final simplification 88.3%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

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(m * Float64(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[(m * N[(N[(1.0 - m), $MachinePrecision] / 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 87.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 99.0%

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

   6. Taylor expanded in v around 0 99.9%

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

   7. Taylor expanded in m around 0 98.1%

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

   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 0.1%

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

      1. pow1 0.1%

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

      2. *-commutative 0.1%

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

      3. sub-neg 0.1%

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

      4. metadata-eval 0.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 76.5%

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

      7. sqr-neg 76.5%

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

      8. sqrt-unprod 76.5%

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

      9. add-sqr-sqrt 76.5%

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

      10. neg-sub0 76.5%

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

      11. associate--r- 76.5%

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

      12. metadata-eval 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. unpow1 76.5%

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

      2. +-commutative 76.5%

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

   7. Simplified 76.5%

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

   8. Taylor expanded in v around 0 76.5%

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

      1. sub-neg 76.5%

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

      2. metadata-eval 76.5%

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

      3. associate-*r/ 76.5%

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

   10. Simplified 76.5%

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

4. Final simplification 88.3%

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

Alternative 7: 62.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-150}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 6e-150) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6e-150) {
		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 <= 6d-150) 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 <= 6e-150) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6e-150:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6e-150)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6e-150)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6e-150], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.0000000000000003e-150

   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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 72.6%

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

   if 6.0000000000000003e-150 < 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 36.3%

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

   4. Taylor expanded in v around 0 36.3%

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

      1. mul-1-neg 36.3%

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

      2. unsub-neg 36.3%

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

   6. Simplified 36.3%

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

   7. Taylor expanded in v around 0 28.2%

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

      1. associate-/l* 28.1%

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

   9. Simplified 28.1%

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

   10. Taylor expanded in m around 0 59.7%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-150}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-53}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 7.2e-53) -1.0 1.0))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.2e-53) {
		tmp = -1.0;
	} else {
		tmp = 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 <= 7.2d-53) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7.2e-53) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.2e-53:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.2e-53)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7.2e-53)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.2e-53], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if m < 7.1999999999999998e-53

   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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 54.6%

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

   if 7.1999999999999998e-53 < 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 14.1%

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

      1. pow1 14.1%

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

      2. *-commutative 14.1%

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

      3. sub-neg 14.1%

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

      4. metadata-eval 14.1%

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

      5. add-sqr-sqrt 14.1%

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

      6. sqrt-unprod 78.4%

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

      7. sqr-neg 78.4%

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

      8. sqrt-unprod 64.3%

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

      9. add-sqr-sqrt 64.5%

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

      10. neg-sub0 64.5%

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

      11. associate--r- 64.5%

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

      12. metadata-eval 64.5%

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

   5. Applied egg-rr 64.5%

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

      1. unpow1 64.5%

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

      2. +-commutative 64.5%

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

   7. Simplified 64.5%

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

   8. Taylor expanded in m around 0 3.7%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-53}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.8% 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 99.4%

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

6. Taylor expanded in v around 0 99.9%

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

7. Taylor expanded in m around 0 76.5%

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

8. Final simplification 76.5%

   \[\leadsto \frac{m}{v} + -1 \]
9. Add Preprocessing

Alternative 10: 27.5% 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 27.8%

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

   1. neg-mul-1 27.8%

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

   2. sub-neg 27.8%

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

   3. +-commutative 27.8%

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

   4. distribute-neg-in 27.8%

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

   5. remove-double-neg 27.8%

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

   6. metadata-eval 27.8%

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

7. Simplified 27.8%

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

8. Final simplification 27.8%

   \[\leadsto m + -1 \]
9. Add Preprocessing

Alternative 11: 25.1% 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 25.6%

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

6. Final simplification 25.6%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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)))