b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 12.1s

Alternatives: 13

Speedup: 1.0×

• 42.4% 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

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

FPCore

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

C

double code(double m, double v) {
	return (((m - (m * 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 - (m * m)) / v) + (-1.0d0)) * (1.0d0 - m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   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%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 71.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.4 \cdot 10^{-224}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 3.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 6 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.8 \cdot 10^{-127}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{-115}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.18 \cdot 10^{-12}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.4e-224)
   -1.0
   (if (<= m 3.4e-155)
     (/ m v)
     (if (<= m 6e-153)
       -1.0
       (if (<= m 1.8e-127)
         (/ m v)
         (if (<= m 1.75e-115)
           -1.0
           (if (<= m 1.18e-12) (/ m v) (* m (/ m v)))))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.4e-224) {
		tmp = -1.0;
	} else if (m <= 3.4e-155) {
		tmp = m / v;
	} else if (m <= 6e-153) {
		tmp = -1.0;
	} else if (m <= 1.8e-127) {
		tmp = m / v;
	} else if (m <= 1.75e-115) {
		tmp = -1.0;
	} else if (m <= 1.18e-12) {
		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 <= 6.4d-224) then
        tmp = -1.0d0
    else if (m <= 3.4d-155) then
        tmp = m / v
    else if (m <= 6d-153) then
        tmp = -1.0d0
    else if (m <= 1.8d-127) then
        tmp = m / v
    else if (m <= 1.75d-115) then
        tmp = -1.0d0
    else if (m <= 1.18d-12) 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 <= 6.4e-224) {
		tmp = -1.0;
	} else if (m <= 3.4e-155) {
		tmp = m / v;
	} else if (m <= 6e-153) {
		tmp = -1.0;
	} else if (m <= 1.8e-127) {
		tmp = m / v;
	} else if (m <= 1.75e-115) {
		tmp = -1.0;
	} else if (m <= 1.18e-12) {
		tmp = m / v;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.4e-224:
		tmp = -1.0
	elif m <= 3.4e-155:
		tmp = m / v
	elif m <= 6e-153:
		tmp = -1.0
	elif m <= 1.8e-127:
		tmp = m / v
	elif m <= 1.75e-115:
		tmp = -1.0
	elif m <= 1.18e-12:
		tmp = m / v
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.4e-224)
		tmp = -1.0;
	elseif (m <= 3.4e-155)
		tmp = Float64(m / v);
	elseif (m <= 6e-153)
		tmp = -1.0;
	elseif (m <= 1.8e-127)
		tmp = Float64(m / v);
	elseif (m <= 1.75e-115)
		tmp = -1.0;
	elseif (m <= 1.18e-12)
		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 <= 6.4e-224)
		tmp = -1.0;
	elseif (m <= 3.4e-155)
		tmp = m / v;
	elseif (m <= 6e-153)
		tmp = -1.0;
	elseif (m <= 1.8e-127)
		tmp = m / v;
	elseif (m <= 1.75e-115)
		tmp = -1.0;
	elseif (m <= 1.18e-12)
		tmp = m / v;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.4e-224], -1.0, If[LessEqual[m, 3.4e-155], N[(m / v), $MachinePrecision], If[LessEqual[m, 6e-153], -1.0, If[LessEqual[m, 1.8e-127], N[(m / v), $MachinePrecision], If[LessEqual[m, 1.75e-115], -1.0, If[LessEqual[m, 1.18e-12], N[(m / v), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if m < 6.3999999999999997e-224 or 3.4e-155 < m < 6e-153 or 1.8e-127 < m < 1.7500000000000001e-115

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 81.1%

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

   if 6.3999999999999997e-224 < m < 3.4e-155 or 6e-153 < m < 1.8e-127 or 1.7500000000000001e-115 < m < 1.18000000000000002e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 99.8%

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

   4. Taylor expanded in v around 0 74.7%

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

   5. Taylor expanded in m around 0 74.6%

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

   if 1.18000000000000002e-12 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 2.3%

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

      1. sub-neg 2.3%

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

      2. distribute-rgt-in 2.3%

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

      3. *-un-lft-identity 2.3%

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

      4. sub-neg 2.3%

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

      5. metadata-eval 2.3%

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

      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 75.6%

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

      8. sqr-neg 75.6%

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

      9. sqrt-unprod 75.6%

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

      10. add-sqr-sqrt 75.6%

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

      11. sub-neg 75.6%

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

      12. metadata-eval 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. distribute-rgt1-in 75.6%

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

      2. +-commutative 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in v around 0 75.6%

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

      1. associate-/l* 75.6%

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

      2. +-commutative 75.6%

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

   10. Simplified 75.6%

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

       1. clear-num 75.6%

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

       2. inv-pow 75.6%

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

       3. +-commutative 75.6%

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

   12. Applied egg-rr 75.6%

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

       1. unpow-1 75.6%

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

       2. +-commutative 75.6%

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

   14. Simplified 75.6%

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

   15. Taylor expanded in m around inf 73.9%

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

   16. Taylor expanded in v around 0 73.9%

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

Alternative 3: 60.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.3 \cdot 10^{-224}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 6.5 \cdot 10^{-155} \lor \neg \left(m \leq 8.2 \cdot 10^{-153}\right) \land \left(m \leq 2.1 \cdot 10^{-127} \lor \neg \left(m \leq 7 \cdot 10^{-115}\right)\right):\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.3e-224)
   -1.0
   (if (or (<= m 6.5e-155)
           (and (not (<= m 8.2e-153))
                (or (<= m 2.1e-127) (not (<= m 7e-115)))))
     (/ m v)
     -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.3e-224) {
		tmp = -1.0;
	} else if ((m <= 6.5e-155) || (!(m <= 8.2e-153) && ((m <= 2.1e-127) || !(m <= 7e-115)))) {
		tmp = m / v;
	} 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 <= 4.3d-224) then
        tmp = -1.0d0
    else if ((m <= 6.5d-155) .or. (.not. (m <= 8.2d-153)) .and. (m <= 2.1d-127) .or. (.not. (m <= 7d-115))) then
        tmp = m / v
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.3e-224) {
		tmp = -1.0;
	} else if ((m <= 6.5e-155) || (!(m <= 8.2e-153) && ((m <= 2.1e-127) || !(m <= 7e-115)))) {
		tmp = m / v;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.3e-224:
		tmp = -1.0
	elif (m <= 6.5e-155) or (not (m <= 8.2e-153) and ((m <= 2.1e-127) or not (m <= 7e-115))):
		tmp = m / v
	else:
		tmp = -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.3e-224)
		tmp = -1.0;
	elseif ((m <= 6.5e-155) || (!(m <= 8.2e-153) && ((m <= 2.1e-127) || !(m <= 7e-115))))
		tmp = Float64(m / v);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.3e-224)
		tmp = -1.0;
	elseif ((m <= 6.5e-155) || (~((m <= 8.2e-153)) && ((m <= 2.1e-127) || ~((m <= 7e-115)))))
		tmp = m / v;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.3e-224], -1.0, If[Or[LessEqual[m, 6.5e-155], And[N[Not[LessEqual[m, 8.2e-153]], $MachinePrecision], Or[LessEqual[m, 2.1e-127], N[Not[LessEqual[m, 7e-115]], $MachinePrecision]]]], N[(m / v), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if m < 4.3e-224 or 6.5e-155 < m < 8.2e-153 or 2.1000000000000001e-127 < m < 7.0000000000000004e-115

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 81.1%

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

   if 4.3e-224 < m < 6.5e-155 or 8.2e-153 < m < 2.1000000000000001e-127 or 7.0000000000000004e-115 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 38.4%

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

   4. Taylor expanded in v around 0 29.1%

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

   5. Taylor expanded in m around 0 61.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.3 \cdot 10^{-224}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 6.5 \cdot 10^{-155} \lor \neg \left(m \leq 8.2 \cdot 10^{-153}\right) \land \left(m \leq 2.1 \cdot 10^{-127} \lor \neg \left(m \leq 7 \cdot 10^{-115}\right)\right):\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;-1 + \frac{m \cdot \left(1 + m \cdot -2\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 - m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.5e-6)
   (+ -1.0 (/ (* m (+ 1.0 (* m -2.0))) v))
   (* (- 1.0 m) (- -1.0 (* m (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.5e-6) {
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	} else {
		tmp = (1.0 - m) * (-1.0 - (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 <= 4.5d-6) then
        tmp = (-1.0d0) + ((m * (1.0d0 + (m * (-2.0d0)))) / v)
    else
        tmp = (1.0d0 - m) * ((-1.0d0) - (m * (m / v)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.5e-6) {
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	} else {
		tmp = (1.0 - m) * (-1.0 - (m * (m / v)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.5e-6:
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v)
	else:
		tmp = (1.0 - m) * (-1.0 - (m * (m / v)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.5e-6)
		tmp = Float64(-1.0 + Float64(Float64(m * Float64(1.0 + Float64(m * -2.0))) / v));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 - Float64(m * Float64(m / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.5e-6)
		tmp = -1.0 + ((m * (1.0 + (m * -2.0))) / v);
	else
		tmp = (1.0 - m) * (-1.0 - (m * (m / v)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.5e-6], N[(-1.0 + N[(N[(m * N[(1.0 + N[(m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 - N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.50000000000000011e-6

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.8%

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

   6. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in m around 0 99.8%

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

       1. *-commutative 99.8%

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

   11. Simplified 99.8%

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

   12. Taylor expanded in v around 0 100.0%

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

   if 4.50000000000000011e-6 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around inf 97.5%

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

      1. neg-mul-1 97.5%

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

      2. distribute-neg-frac2 97.5%

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

   7. Simplified 97.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;-1 + \frac{m \cdot \left(1 + m \cdot -2\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 - m \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

Alternative 7: 86.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-58}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m + 1\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.25e-58) (+ -1.0 (/ m v)) (/ (* m (+ m 1.0)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-58) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m * (m + 1.0)) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.25d-58) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = (m * (m + 1.0d0)) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-58) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m * (m + 1.0)) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.25e-58:
		tmp = -1.0 + (m / v)
	else:
		tmp = (m * (m + 1.0)) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.25e-58)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(Float64(m * Float64(m + 1.0)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.25e-58)
		tmp = -1.0 + (m / v);
	else
		tmp = (m * (m + 1.0)) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.25e-58], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m + 1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.24999999999999994e-58

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.8%

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

   6. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in m around 0 100.0%

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

   if 1.24999999999999994e-58 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 14.5%

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

      1. sub-neg 14.5%

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

      2. distribute-rgt-in 14.5%

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

      3. *-un-lft-identity 14.5%

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

      4. sub-neg 14.5%

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

      5. metadata-eval 14.5%

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

      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 78.6%

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

      8. sqr-neg 78.6%

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

      9. sqrt-unprod 78.6%

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

      10. add-sqr-sqrt 78.6%

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

      11. sub-neg 78.6%

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

      12. metadata-eval 78.6%

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

   5. Applied egg-rr 78.6%

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

      1. distribute-rgt1-in 78.6%

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

      2. +-commutative 78.6%

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

   7. Simplified 78.6%

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

   8. Taylor expanded in v around 0 78.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-58}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m + 1\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-58}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.25e-58) (+ -1.0 (/ m v)) (/ m (/ v (+ m 1.0)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-58) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m / (v / (m + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.25d-58) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = m / (v / (m + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-58) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m / (v / (m + 1.0));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.25e-58:
		tmp = -1.0 + (m / v)
	else:
		tmp = m / (v / (m + 1.0))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.25e-58)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(m / Float64(v / Float64(m + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.25e-58)
		tmp = -1.0 + (m / v);
	else
		tmp = m / (v / (m + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.25e-58], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / N[(m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.24999999999999994e-58

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.8%

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

   6. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in m around 0 100.0%

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

   if 1.24999999999999994e-58 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 14.5%

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

      1. sub-neg 14.5%

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

      2. distribute-rgt-in 14.5%

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

      3. *-un-lft-identity 14.5%

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

      4. sub-neg 14.5%

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

      5. metadata-eval 14.5%

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

      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 78.6%

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

      8. sqr-neg 78.6%

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

      9. sqrt-unprod 78.6%

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

      10. add-sqr-sqrt 78.6%

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

      11. sub-neg 78.6%

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

      12. metadata-eval 78.6%

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

   5. Applied egg-rr 78.6%

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

      1. distribute-rgt1-in 78.6%

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

      2. +-commutative 78.6%

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

   7. Simplified 78.6%

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

   8. Taylor expanded in v around 0 78.6%

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

      1. associate-/l* 78.5%

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

      2. +-commutative 78.5%

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

   10. Simplified 78.5%

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

       1. clear-num 78.5%

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

       2. inv-pow 78.5%

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

       3. +-commutative 78.5%

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

   12. Applied egg-rr 78.5%

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

       1. unpow-1 78.5%

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

       2. +-commutative 78.5%

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

   14. Simplified 78.5%

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

       1. un-div-inv 78.6%

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

   16. Applied egg-rr 78.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-58}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4 \cdot 10^{-63}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4e-63) (+ -1.0 (/ m v)) (* m (/ (+ m 1.0) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4e-63) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 4d-63) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = m * ((m + 1.0d0) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4e-63) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4e-63:
		tmp = -1.0 + (m / v)
	else:
		tmp = m * ((m + 1.0) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4e-63)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(m * Float64(Float64(m + 1.0) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4e-63)
		tmp = -1.0 + (m / v);
	else
		tmp = m * ((m + 1.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4e-63], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m + 1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.00000000000000027e-63

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.8%

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

   6. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in m around 0 100.0%

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

   if 4.00000000000000027e-63 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 15.6%

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

      1. sub-neg 15.6%

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

      2. distribute-rgt-in 15.6%

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

      3. *-un-lft-identity 15.6%

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

      4. sub-neg 15.6%

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

      5. metadata-eval 15.6%

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

      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 78.9%

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

      8. sqr-neg 78.9%

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

      9. sqrt-unprod 78.9%

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

      10. add-sqr-sqrt 78.9%

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

      11. sub-neg 78.9%

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

      12. metadata-eval 78.9%

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

   5. Applied egg-rr 78.9%

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

      1. distribute-rgt1-in 78.9%

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

      2. +-commutative 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in v around 0 78.2%

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

      1. associate-/l* 78.2%

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

      2. +-commutative 78.2%

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

   10. Simplified 78.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4 \cdot 10^{-63}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.18 \cdot 10^{-12}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.18e-12) (+ -1.0 (/ m v)) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.18e-12) {
		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 <= 1.18d-12) 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 <= 1.18e-12) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.18e-12:
		tmp = -1.0 + (m / v)
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.18e-12)
		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 <= 1.18e-12)
		tmp = -1.0 + (m / v);
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.18e-12], 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 < 1.18000000000000002e-12

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.8%

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

   6. Taylor expanded in v around 0 100.0%

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

      1. associate-/l* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in m around 0 99.9%

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

   if 1.18000000000000002e-12 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 2.3%

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

      1. sub-neg 2.3%

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

      2. distribute-rgt-in 2.3%

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

      3. *-un-lft-identity 2.3%

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

      4. sub-neg 2.3%

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

      5. metadata-eval 2.3%

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

      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 75.6%

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

      8. sqr-neg 75.6%

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

      9. sqrt-unprod 75.6%

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

      10. add-sqr-sqrt 75.6%

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

      11. sub-neg 75.6%

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

      12. metadata-eval 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. distribute-rgt1-in 75.6%

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

      2. +-commutative 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in v around 0 75.6%

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

      1. associate-/l* 75.6%

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

      2. +-commutative 75.6%

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

   10. Simplified 75.6%

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

       1. clear-num 75.6%

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

       2. inv-pow 75.6%

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

       3. +-commutative 75.6%

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

   12. Applied egg-rr 75.6%

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

       1. unpow-1 75.6%

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

       2. +-commutative 75.6%

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

   14. Simplified 75.6%

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

   15. Taylor expanded in m around inf 73.9%

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

   16. Taylor expanded in v around 0 73.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.18 \cdot 10^{-12}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0 49.2%

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

   1. sub-neg 49.2%

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

   2. distribute-rgt-in 49.2%

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

   3. *-un-lft-identity 49.2%

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

   4. sub-neg 49.2%

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

   5. metadata-eval 49.2%

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

   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 87.3%

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

   8. sqr-neg 87.3%

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

   9. sqrt-unprod 87.3%

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

   10. add-sqr-sqrt 87.3%

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

   11. sub-neg 87.3%

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

   12. metadata-eval 87.3%

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

5. Applied egg-rr 87.3%

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

   1. distribute-rgt1-in 87.3%

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

   2. +-commutative 87.3%

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

7. Simplified 87.3%

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

Alternative 12: 27.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 24.9%

   \[\leadsto \color{blue}{-1 \cdot \left(1 - m\right)} \]
6. 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%

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

7. Simplified 24.9%

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

8. Final simplification 24.9%

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

Alternative 13: 24.7% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 22.4%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))