b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.0s

Alternatives: 11

Speedup: 1.0×

• 41.2% 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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.08 \cdot 10^{-61}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m - m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.08e-61) (+ -1.0 (+ m (/ m v))) (/ (- m (* m m)) (/ v (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.08e-61) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m - (m * m)) / (v / (1.0 - 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 <= 1.08d-61) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m - (m * m)) / (v / (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.08e-61) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m - (m * m)) / (v / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.08e-61:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m - (m * m)) / (v / (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.08e-61)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m - Float64(m * m)) / Float64(v / Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.08e-61)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m - (m * m)) / (v / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.08e-61], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.08000000000000008e-61

   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 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

   6. Simplified 100.0%

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

   if 1.08000000000000008e-61 < m

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. mul-1-neg 99.9%

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

      5. sub-neg 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.08 \cdot 10^{-61}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m - m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 3: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (+ -1.0 (+ m (/ m v))) (* (* m (/ m v)) (+ m -2.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (m / v)) * (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 <= 2.4d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m * (m / v)) * (m + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (m / v)) * (m + -2.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m * (m / v)) * (m + -2.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(m / v)) * Float64(m + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m * (m / v)) * (m + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(m + -2.0), $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 98.0%

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

      1. sub-neg 98.0%

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

      2. metadata-eval 98.0%

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

      3. +-commutative 98.0%

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

      4. *-commutative 98.0%

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

      5. distribute-lft-in 98.0%

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

      6. *-rgt-identity 98.0%

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

      7. associate-*r/ 98.3%

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

      8. *-rgt-identity 98.3%

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

   6. Simplified 98.3%

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

   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. Step-by-step derivation

      1. flip-- 99.9%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in m around inf 21.8%

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

      1. unpow2 21.8%

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

      2. cube-mult 21.7%

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

      3. associate-*r/ 21.7%

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

      4. distribute-rgt-out 99.1%

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

      5. associate-*r/ 99.1%

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

   8. Simplified 99.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \]

Alternative 4: 72.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.25e-198) -1.0 (if (<= m 2.2) (/ m v) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-198) {
		tmp = -1.0;
	} else if (m <= 2.2) {
		tmp = m / v;
	} 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 <= 1.25d-198) then
        tmp = -1.0d0
    else if (m <= 2.2d0) then
        tmp = m / v
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-198) {
		tmp = -1.0;
	} else if (m <= 2.2) {
		tmp = m / v;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.25e-198:
		tmp = -1.0
	elif m <= 2.2:
		tmp = m / v
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.25e-198)
		tmp = -1.0;
	elseif (m <= 2.2)
		tmp = Float64(m / v);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.25e-198)
		tmp = -1.0;
	elseif (m <= 2.2)
		tmp = m / v;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.25e-198], -1.0, If[LessEqual[m, 2.2], N[(m / v), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.25e-198

   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 87.4%

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

   if 1.25e-198 < m < 2.2000000000000002

   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 97.4%

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

      1. sub-neg 97.4%

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

      2. distribute-rgt-in 97.4%

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

      3. *-un-lft-identity 97.4%

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

      4. sub-neg 97.4%

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

      5. metadata-eval 97.4%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 97.4%

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

      8. sqr-neg 97.4%

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

      9. sqrt-unprod 97.4%

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

      10. add-sqr-sqrt 97.4%

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

      11. sub-neg 97.4%

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

      12. metadata-eval 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. distribute-rgt1-in 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in v around 0 68.8%

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

   8. Taylor expanded in m around 0 68.9%

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

   if 2.2000000000000002 < 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. sub-neg 0.1%

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

      2. distribute-rgt-in 0.1%

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

      3. *-un-lft-identity 0.1%

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

      4. sub-neg 0.1%

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

      5. metadata-eval 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 79.4%

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

      8. sqr-neg 79.4%

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

      9. sqrt-unprod 79.4%

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

      10. add-sqr-sqrt 79.4%

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

      11. sub-neg 79.4%

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

      12. metadata-eval 79.4%

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

   4. Applied egg-rr 79.4%

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

      1. distribute-rgt1-in 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in m around inf 79.4%

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

      1. unpow2 79.4%

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

      2. associate-*r/ 79.4%

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

   9. Simplified 79.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 5: 72.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.15 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(1 + m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.15e-198) -1.0 (* (/ m v) (+ 1.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.15e-198) {
		tmp = -1.0;
	} else {
		tmp = (m / v) * (1.0 + 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 <= 1.15d-198) then
        tmp = -1.0d0
    else
        tmp = (m / v) * (1.0d0 + m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.15e-198) {
		tmp = -1.0;
	} else {
		tmp = (m / v) * (1.0 + m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.15e-198:
		tmp = -1.0
	else:
		tmp = (m / v) * (1.0 + m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.15e-198)
		tmp = -1.0;
	else
		tmp = Float64(Float64(m / v) * Float64(1.0 + m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.15e-198)
		tmp = -1.0;
	else
		tmp = (m / v) * (1.0 + m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.15e-198], -1.0, N[(N[(m / v), $MachinePrecision] * N[(1.0 + m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.15000000000000007e-198

   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 87.4%

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

   if 1.15000000000000007e-198 < 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 35.9%

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

      1. sub-neg 35.9%

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

      2. distribute-rgt-in 35.9%

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

      3. *-un-lft-identity 35.9%

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

      4. sub-neg 35.9%

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

      5. metadata-eval 35.9%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 86.0%

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

      8. sqr-neg 86.0%

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

      9. sqrt-unprod 86.0%

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

      10. add-sqr-sqrt 86.0%

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

      11. sub-neg 86.0%

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

      12. metadata-eval 86.0%

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

   4. Applied egg-rr 86.0%

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

      1. distribute-rgt1-in 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. 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%

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

      2. +-commutative 75.5%

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

   9. Simplified 75.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.15 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(1 + m\right)\\ \end{array} \]

Alternative 6: 87.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(1.0 + m), $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 98.0%

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

      1. sub-neg 98.0%

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

      2. metadata-eval 98.0%

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

      3. +-commutative 98.0%

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

      4. *-commutative 98.0%

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

      5. distribute-lft-in 98.0%

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

      6. *-rgt-identity 98.0%

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

      7. associate-*r/ 98.3%

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

      8. *-rgt-identity 98.3%

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

   6. Simplified 98.3%

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

   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. sub-neg 0.1%

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

      2. distribute-rgt-in 0.1%

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

      3. *-un-lft-identity 0.1%

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

      4. sub-neg 0.1%

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

      5. metadata-eval 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 79.4%

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

      8. sqr-neg 79.4%

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

      9. sqrt-unprod 79.4%

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

      10. add-sqr-sqrt 79.4%

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

      11. sub-neg 79.4%

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

      12. metadata-eval 79.4%

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

   4. Applied egg-rr 79.4%

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

      1. distribute-rgt1-in 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in v around 0 79.4%

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

      1. associate-*l/ 79.4%

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

      2. +-commutative 79.4%

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

   9. Simplified 79.4%

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

4. Final simplification 88.2%

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

Alternative 7: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6) (+ -1.0 (+ m (/ m v))) (/ (* m m) (/ v m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * m) / (v / m);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.6d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m * m) / (v / m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * m) / (v / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m * m) / (v / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m * m) / Float64(v / m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m * m) / (v / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * m), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000009

   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 98.0%

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

      1. sub-neg 98.0%

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

      2. metadata-eval 98.0%

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

      3. +-commutative 98.0%

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

      4. *-commutative 98.0%

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

      5. distribute-lft-in 98.0%

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

      6. *-rgt-identity 98.0%

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

      7. associate-*r/ 98.3%

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

      8. *-rgt-identity 98.3%

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

   6. Simplified 98.3%

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

   if 2.60000000000000009 < 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 98.5%

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

      1. unpow3 98.5%

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

      2. *-un-lft-identity 98.5%

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

      3. times-frac 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. associate-*l/ 98.5%

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

      2. associate-/l* 98.5%

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

      3. clear-num 98.5%

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

   8. Applied egg-rr 98.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]

Alternative 8: 61.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-199}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 6e-199) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6e-199) {
		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-199) 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-199) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6e-199:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6e-199)
		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-199)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6e-199], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.99999999999999966e-199

   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 87.4%

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

   if 5.99999999999999966e-199 < 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 35.9%

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

      1. sub-neg 35.9%

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

      2. distribute-rgt-in 35.9%

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

      3. *-un-lft-identity 35.9%

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

      4. sub-neg 35.9%

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

      5. metadata-eval 35.9%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 86.0%

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

      8. sqr-neg 86.0%

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

      9. sqrt-unprod 86.0%

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

      10. add-sqr-sqrt 86.0%

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

      11. sub-neg 86.0%

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

      12. metadata-eval 86.0%

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

   4. Applied egg-rr 86.0%

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

      1. distribute-rgt1-in 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. Taylor expanded in v around 0 75.5%

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

   8. Taylor expanded in m around 0 62.5%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6 \cdot 10^{-199}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 9: 27.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 10^{-61}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1e-61) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1e-61) {
		tmp = -1.0;
	} else {
		tmp = 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 <= 1d-61) then
        tmp = -1.0d0
    else
        tmp = m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1e-61) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1e-61:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1e-61)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1e-61)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1e-61], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 1e-61

   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 54.0%

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

   if 1e-61 < m

   1. Initial program 99.8%

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

      1. flip-- 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in m around inf 30.5%

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

      1. +-commutative 30.5%

         \[\leadsto -2 \cdot \frac{{m}^{2}}{v} + \color{blue}{\left(\frac{{m}^{3}}{v} + \left(1 + \frac{1}{v}\right) \cdot m\right)} \]

      2. associate-+r+ 30.6%

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}\right) + \left(1 + \frac{1}{v}\right) \cdot m} \]

      3. unpow2 30.6%

         \[\leadsto \left(-2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v}\right) + \left(1 + \frac{1}{v}\right) \cdot m \]

      4. cube-mult 30.5%

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

      5. associate-*r/ 30.5%

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

      6. distribute-rgt-out 99.8%

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

      7. associate-*r/ 99.9%

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

      8. *-commutative 99.9%

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

      9. distribute-lft-in 99.9%

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

      10. *-rgt-identity 99.9%

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

      11. associate-*r/ 99.9%

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

      12. *-rgt-identity 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in v around inf 5.8%

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

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 10^{-61}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]

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/ 99.9%

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

   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 24.9%

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

   1. neg-mul-1 24.9%

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

   2. neg-sub0 24.9%

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

   3. associate--r- 24.9%

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

   4. metadata-eval 24.9%

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

6. Simplified 24.9%

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

7. Final simplification 24.9%

   \[\leadsto m + -1 \]

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/ 99.9%

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

   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 22.2%

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

5. Final simplification 22.2%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023178 
(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)))