b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 5.7s

Alternatives: 11

Speedup: 1.0×

• 42.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $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* 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-*l/ 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. Final simplification 100.0%

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

Alternative 3: 87.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.32:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.32) (* (- 1.0 m) (+ -1.0 (/ m v))) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.32) {
		tmp = (1.0 - m) * (-1.0 + (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 <= 0.32d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (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 <= 0.32) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.32:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.32)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + 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 <= 0.32)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.32], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.320000000000000007

   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 99.3%

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

   if 0.320000000000000007 < 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 0 0.2%

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

      1. +-commutative 0.2%

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

      2. distribute-rgt-in 0.2%

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

      3. neg-mul-1 0.2%

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

      4. neg-sub0 0.2%

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

      5. metadata-eval 0.2%

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

      6. associate--r- 0.2%

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

      7. metadata-eval 0.2%

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

      8. metadata-eval 0.2%

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

      9. add-sqr-sqrt 0.2%

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

      10. sqrt-unprod 79.2%

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

      11. sqr-neg 79.2%

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

      12. sqrt-unprod 79.1%

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

      13. add-sqr-sqrt 79.1%

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

      14. neg-sub0 79.1%

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

      15. metadata-eval 79.1%

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

      16. associate--r- 79.1%

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

      17. metadata-eval 79.1%

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

      18. metadata-eval 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. distribute-rgt1-in 79.1%

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

      2. +-commutative 79.1%

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

   8. Simplified 79.1%

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

   9. Taylor expanded in m around inf 79.3%

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

       1. unpow2 79.3%

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

       2. associate-*r/ 79.3%

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

   11. Simplified 79.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.32:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 4: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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 98.7%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 98.3%

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

      1. associate-*r/ 98.3%

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

      2. neg-mul-1 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in v around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. associate-/l* 98.3%

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

      3. distribute-neg-frac 98.3%

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

      4. unpow2 98.3%

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

      5. distribute-rgt-neg-in 98.3%

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

   9. Simplified 98.3%

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

       1. associate-/r/ 98.2%

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

       2. *-commutative 98.2%

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

       3. associate-/l* 98.3%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 0.0%

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

       6. sqr-neg 0.0%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 0.0%

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

       9. sub-neg 0.0%

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

       10. add-sqr-sqrt 0.0%

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

       11. sqrt-unprod 98.2%

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

       12. sqr-neg 98.2%

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

       13. sqrt-unprod 98.2%

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

       14. add-sqr-sqrt 98.2%

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

       15. +-commutative 98.2%

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

   11. Applied egg-rr 98.2%

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

4. Final simplification 98.5%

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

Alternative 5: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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 98.7%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 98.3%

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

      1. associate-*r/ 98.3%

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

      2. neg-mul-1 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in v around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. associate-/l* 98.3%

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

      3. distribute-neg-frac 98.3%

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

      4. unpow2 98.3%

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

      5. distribute-rgt-neg-in 98.3%

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

   9. Simplified 98.3%

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

   10. Taylor expanded in m around 0 20.6%

       \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   11. Step-by-step derivation

       1. +-commutative 20.6%

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

       2. mul-1-neg 20.6%

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

       3. unpow2 20.6%

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

       4. associate-/l* 20.6%

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

       5. unsub-neg 20.6%

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

       6. cube-mult 20.6%

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

       7. associate-/l* 20.6%

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

       8. associate-/l* 20.6%

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

       9. *-lft-identity 20.6%

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

       10. associate-*l/ 20.6%

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

       11. associate-/r/ 20.6%

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

       12. div-sub 98.3%

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

       13. associate-/r/ 98.3%

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

       14. sub-neg 98.3%

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

       15. metadata-eval 98.3%

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

   12. Simplified 98.3%

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

4. Final simplification 98.5%

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

Alternative 6: 73.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4 \cdot 10^{-126}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4e-126) -1.0 (if (<= m 0.27) (/ m v) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4e-126) {
		tmp = -1.0;
	} else if (m <= 0.27) {
		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 <= 2.4d-126) then
        tmp = -1.0d0
    else if (m <= 0.27d0) 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 <= 2.4e-126) {
		tmp = -1.0;
	} else if (m <= 0.27) {
		tmp = m / v;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4e-126:
		tmp = -1.0
	elif m <= 0.27:
		tmp = m / v
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4e-126)
		tmp = -1.0;
	elseif (m <= 0.27)
		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 <= 2.4e-126)
		tmp = -1.0;
	elseif (m <= 0.27)
		tmp = m / v;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4e-126], -1.0, If[LessEqual[m, 0.27], N[(m / v), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.40000000000000007e-126

   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 74.5%

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

   if 2.40000000000000007e-126 < m < 0.27000000000000002

   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 98.1%

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

   5. Taylor expanded in v around 0 78.5%

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

      1. *-commutative 78.5%

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

      2. sub-neg 78.5%

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

      3. metadata-eval 78.5%

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

      4. distribute-neg-in 78.5%

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

      5. neg-mul-1 78.5%

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

      6. *-commutative 78.5%

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

      7. associate-*r/ 78.5%

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

      8. *-commutative 78.5%

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

      9. associate-*r* 78.5%

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

      10. mul-1-neg 78.5%

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

      11. *-commutative 78.5%

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

      12. associate-*l/ 78.5%

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

      13. associate-*r/ 78.2%

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

      14. distribute-rgt-neg-in 78.2%

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

      15. distribute-neg-frac 78.2%

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

      16. distribute-neg-in 78.2%

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

      17. metadata-eval 78.2%

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

      18. sub-neg 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in m around 0 78.5%

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

   if 0.27000000000000002 < 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 0 0.2%

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

      1. +-commutative 0.2%

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

      2. distribute-rgt-in 0.2%

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

      3. neg-mul-1 0.2%

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

      4. neg-sub0 0.2%

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

      5. metadata-eval 0.2%

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

      6. associate--r- 0.2%

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

      7. metadata-eval 0.2%

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

      8. metadata-eval 0.2%

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

      9. add-sqr-sqrt 0.2%

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

      10. sqrt-unprod 79.2%

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

      11. sqr-neg 79.2%

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

      12. sqrt-unprod 79.1%

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

      13. add-sqr-sqrt 79.1%

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

      14. neg-sub0 79.1%

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

      15. metadata-eval 79.1%

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

      16. associate--r- 79.1%

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

      17. metadata-eval 79.1%

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

      18. metadata-eval 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. distribute-rgt1-in 79.1%

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

      2. +-commutative 79.1%

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

   8. Simplified 79.1%

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

   9. Taylor expanded in m around inf 79.3%

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

       1. unpow2 79.3%

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

       2. associate-*r/ 79.3%

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

   11. Simplified 79.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4 \cdot 10^{-126}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 7: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.27:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.27) (+ -1.0 (/ m v)) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.27) {
		tmp = -1.0 + (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 <= 0.27d0) then
        tmp = (-1.0d0) + (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 <= 0.27) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.27:
		tmp = -1.0 + (m / v)
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.27)
		tmp = Float64(-1.0 + 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 <= 0.27)
		tmp = -1.0 + (m / v);
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.27], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.27000000000000002

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. associate-/r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. neg-mul-1 100.0%

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

      5. clear-num 99.8%

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

      6. associate-/r/ 99.8%

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

      7. clear-num 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r* 99.8%

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

      10. *-commutative 99.8%

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

      11. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in m around 0 99.0%

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

      1. sub-neg 99.0%

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

      2. *-commutative 99.0%

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

      3. distribute-rgt-in 99.0%

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

      4. *-lft-identity 99.0%

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

      5. associate-*l/ 99.3%

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

      6. *-lft-identity 99.3%

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

      7. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in v around 0 99.3%

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

   if 0.27000000000000002 < 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 0 0.2%

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

      1. +-commutative 0.2%

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

      2. distribute-rgt-in 0.2%

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

      3. neg-mul-1 0.2%

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

      4. neg-sub0 0.2%

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

      5. metadata-eval 0.2%

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

      6. associate--r- 0.2%

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

      7. metadata-eval 0.2%

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

      8. metadata-eval 0.2%

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

      9. add-sqr-sqrt 0.2%

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

      10. sqrt-unprod 79.2%

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

      11. sqr-neg 79.2%

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

      12. sqrt-unprod 79.1%

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

      13. add-sqr-sqrt 79.1%

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

      14. neg-sub0 79.1%

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

      15. metadata-eval 79.1%

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

      16. associate--r- 79.1%

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

      17. metadata-eval 79.1%

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

      18. metadata-eval 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. distribute-rgt1-in 79.1%

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

      2. +-commutative 79.1%

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

   8. Simplified 79.1%

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

   9. Taylor expanded in m around inf 79.3%

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

       1. unpow2 79.3%

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

       2. associate-*r/ 79.3%

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

   11. Simplified 79.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.27:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 8: 61.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.35 \cdot 10^{-126}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.35e-126) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.35e-126) {
		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 <= 2.35d-126) 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 <= 2.35e-126) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.35e-126:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.35e-126)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.35e-126)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.35e-126], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.35000000000000009e-126

   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 74.5%

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

   if 2.35000000000000009e-126 < 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 0 28.7%

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

   5. Taylor expanded in v around 0 23.0%

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

      1. *-commutative 23.0%

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

      2. sub-neg 23.0%

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

      3. metadata-eval 23.0%

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

      4. distribute-neg-in 23.0%

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

      5. neg-mul-1 23.0%

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

      6. *-commutative 23.0%

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

      7. associate-*r/ 23.0%

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

      8. *-commutative 23.0%

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

      9. associate-*r* 23.0%

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

      10. mul-1-neg 23.0%

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

      11. *-commutative 23.0%

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

      12. associate-*l/ 23.0%

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

      13. associate-*r/ 22.9%

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

      14. distribute-rgt-neg-in 22.9%

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

      15. distribute-neg-frac 22.9%

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

      16. distribute-neg-in 22.9%

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

      17. metadata-eval 22.9%

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

      18. sub-neg 22.9%

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

   7. Simplified 22.9%

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

   8. Taylor expanded in m around 0 63.6%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.35 \cdot 10^{-126}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 9: 24.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4 \cdot 10^{-56}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.4e-56) -1.0 1.0))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4e-56) {
		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 <= 2.4d-56) 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 <= 2.4e-56) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4e-56:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4e-56)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4e-56)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4e-56], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if m < 2.40000000000000001e-56

   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 61.2%

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

   if 2.40000000000000001e-56 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 83.3%

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

      1. associate-*r/ 83.3%

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

      2. neg-mul-1 83.3%

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

   6. Simplified 83.3%

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

      1. distribute-lft-in 83.3%

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

      2. clear-num 83.3%

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

      3. un-div-inv 83.3%

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

      4. add-sqr-sqrt 0.1%

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

      5. sqrt-unprod 0.2%

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

      6. sqr-neg 0.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 1.3%

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

      9. neg-sub0 1.3%

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

      10. metadata-eval 1.3%

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

      11. associate--r- 1.3%

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

      12. metadata-eval 1.3%

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

      13. metadata-eval 1.3%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 81.6%

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

      16. sqr-neg 81.6%

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

      17. sqrt-unprod 83.3%

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

      18. add-sqr-sqrt 83.3%

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

   8. Applied egg-rr 83.5%

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

   9. Taylor expanded in m around 0 3.6%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4 \cdot 10^{-56}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 26.4% 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/ 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 v around inf 29.7%

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

   1. neg-mul-1 29.7%

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

   2. neg-sub0 29.7%

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

   3. associate--r- 29.7%

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

   4. metadata-eval 29.7%

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

6. Simplified 29.7%

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

7. Final simplification 29.7%

   \[\leadsto m + -1 \]

Alternative 11: 23.9% 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/ 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 27.4%

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

5. Final simplification 27.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023222 
(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)))