b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

Alternatives: 13

Speedup: N/A×

• 41.5% 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* 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. Step-by-step derivation

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.2 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.2e-173)
   -1.0
   (if (<= m 1.0) (* (- 1.0 m) (/ m v)) (* (/ m v) (+ m -1.0)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.2e-173) {
		tmp = -1.0;
	} else if (m <= 1.0) {
		tmp = (1.0 - m) * (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 <= 2.2d-173) then
        tmp = -1.0d0
    else if (m <= 1.0d0) then
        tmp = (1.0d0 - m) * (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 <= 2.2e-173) {
		tmp = -1.0;
	} else if (m <= 1.0) {
		tmp = (1.0 - m) * (m / v);
	} else {
		tmp = (m / v) * (m + -1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.2e-173:
		tmp = -1.0
	elif m <= 1.0:
		tmp = (1.0 - m) * (m / v)
	else:
		tmp = (m / v) * (m + -1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.2e-173)
		tmp = -1.0;
	elseif (m <= 1.0)
		tmp = Float64(Float64(1.0 - m) * Float64(m / v));
	else
		tmp = Float64(Float64(m / v) * Float64(m + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.2e-173)
		tmp = -1.0;
	elseif (m <= 1.0)
		tmp = (1.0 - m) * (m / v);
	else
		tmp = (m / v) * (m + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.2e-173], -1.0, If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.1999999999999999e-173

   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 75.2%

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

   if 2.1999999999999999e-173 < m < 1

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. 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 100.0%

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

      3. *-un-lft-identity 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in v around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-/l* 75.7%

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

      3. mul-1-neg 75.7%

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

      4. *-rgt-identity 75.7%

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

      5. unpow2 75.7%

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

      6. distribute-rgt-neg-in 75.7%

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

      7. distribute-lft-in 75.6%

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

      8. sub-neg 75.6%

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

      9. associate-/l* 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in m around 0 71.7%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. 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. Taylor expanded in v around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.9%

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

      3. mul-1-neg 99.9%

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

      4. *-rgt-identity 99.9%

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

      5. unpow2 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. distribute-lft-in 99.9%

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

      8. sub-neg 99.9%

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

      9. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around 0 0.1%

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

      1. associate-*r/ 0.1%

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

      2. *-commutative 0.1%

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

      3. clear-num 0.1%

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

      4. frac-2neg 0.1%

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

      5. metadata-eval 0.1%

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

      6. distribute-neg-frac 0.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 70.4%

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

      9. sqr-neg 70.4%

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

      10. sqrt-unprod 69.8%

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

      11. add-sqr-sqrt 69.8%

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

      12. *-commutative 69.8%

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

      13. associate-/r* 69.8%

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

   10. Applied egg-rr 69.8%

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

       1. associate-/r/ 69.8%

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

       2. associate-*l/ 69.8%

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

       3. neg-mul-1 69.8%

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

       4. distribute-frac-neg 69.8%

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

       5. associate-/r/ 69.8%

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

       6. *-commutative 69.8%

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

       7. distribute-lft-neg-out 69.8%

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

       8. neg-sub0 69.8%

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

       9. associate--r- 69.8%

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

       10. metadata-eval 69.8%

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

       11. +-commutative 69.8%

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

       12. *-commutative 69.8%

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

       13. +-commutative 69.8%

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

   12. Simplified 69.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.2 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4 \cdot 10^{-22}:\\ \;\;\;\;\left(1 + m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4e-22)
   (* (+ 1.0 m) (+ -1.0 (/ m v)))
   (* m (* (- 1.0 m) (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4e-22) {
		tmp = (1.0 + m) * (-1.0 + (m / v));
	} else {
		tmp = m * ((1.0 - m) * ((1.0 - 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 <= 4d-22) then
        tmp = (1.0d0 + m) * ((-1.0d0) + (m / v))
    else
        tmp = m * ((1.0d0 - m) * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4e-22) {
		tmp = (1.0 + m) * (-1.0 + (m / v));
	} else {
		tmp = m * ((1.0 - m) * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4e-22:
		tmp = (1.0 + m) * (-1.0 + (m / v))
	else:
		tmp = m * ((1.0 - m) * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4e-22)
		tmp = Float64(Float64(1.0 + m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(Float64(1.0 - m) * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4e-22)
		tmp = (1.0 + m) * (-1.0 + (m / v));
	else
		tmp = m * ((1.0 - m) * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4e-22], N[(N[(1.0 + m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(1.0 - m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.0000000000000002e-22

   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 0 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      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 100.0%

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

      8. sqr-neg 100.0%

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

      9. sqrt-unprod 100.0%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   7. Simplified 100.0%

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

   if 4.0000000000000002e-22 < 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. 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. Taylor expanded in v around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.8%

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

      3. mul-1-neg 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. unpow2 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. distribute-lft-in 99.8%

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

      8. sub-neg 99.8%

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

      9. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-/l* 99.8%

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

       2. *-commutative 99.8%

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

       3. div-inv 99.8%

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

       4. clear-num 99.8%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4 \cdot 10^{-22}:\\ \;\;\;\;\left(1 + m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 10^{-26}:\\ \;\;\;\;\left(1 + m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} \cdot \left(m + -1\right)\right) \cdot \left(m + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1e-26)
   (* (+ 1.0 m) (+ -1.0 (/ m v)))
   (* (* (/ m v) (+ m -1.0)) (+ m -1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1e-26) {
		tmp = (1.0 + m) * (-1.0 + (m / v));
	} else {
		tmp = ((m / v) * (m + -1.0)) * (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 <= 1d-26) then
        tmp = (1.0d0 + m) * ((-1.0d0) + (m / v))
    else
        tmp = ((m / v) * (m + (-1.0d0))) * (m + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1e-26) {
		tmp = (1.0 + m) * (-1.0 + (m / v));
	} else {
		tmp = ((m / v) * (m + -1.0)) * (m + -1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1e-26:
		tmp = (1.0 + m) * (-1.0 + (m / v))
	else:
		tmp = ((m / v) * (m + -1.0)) * (m + -1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1e-26)
		tmp = Float64(Float64(1.0 + m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(Float64(m / v) * Float64(m + -1.0)) * Float64(m + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1e-26)
		tmp = (1.0 + m) * (-1.0 + (m / v));
	else
		tmp = ((m / v) * (m + -1.0)) * (m + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1e-26], N[(N[(1.0 + m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m / v), $MachinePrecision] * N[(m + -1.0), $MachinePrecision]), $MachinePrecision] * N[(m + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1e-26

   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 0 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      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 100.0%

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

      8. sqr-neg 100.0%

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

      9. sqrt-unprod 100.0%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   7. Simplified 100.0%

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

   if 1e-26 < 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. 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. Taylor expanded in v around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.8%

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

      3. mul-1-neg 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. unpow2 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. distribute-lft-in 99.8%

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

      8. sub-neg 99.8%

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

      9. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-/r/ 99.8%

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

       2. *-commutative 99.8%

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

       3. associate-*r/ 99.8%

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

       4. associate-*l* 99.8%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 10^{-26}:\\ \;\;\;\;\left(1 + m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{m}{v} \cdot \left(m + -1\right)\right) \cdot \left(m + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-28}:\\ \;\;\;\;\left(1 + m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-28)
   (* (+ 1.0 m) (+ -1.0 (/ m v)))
   (/ (* m (- 1.0 m)) (/ v (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-28) {
		tmp = (1.0 + m) * (-1.0 + (m / v));
	} else {
		tmp = (m * (1.0 - m)) / (v / (1.0 - m));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 5d-28) then
        tmp = (1.0d0 + m) * ((-1.0d0) + (m / v))
    else
        tmp = (m * (1.0d0 - m)) / (v / (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5e-28) {
		tmp = (1.0 + m) * (-1.0 + (m / v));
	} else {
		tmp = (m * (1.0 - m)) / (v / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5e-28:
		tmp = (1.0 + m) * (-1.0 + (m / v))
	else:
		tmp = (m * (1.0 - m)) / (v / (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-28)
		tmp = Float64(Float64(1.0 + m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(1.0 - m)) / Float64(v / Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-28)
		tmp = (1.0 + m) * (-1.0 + (m / v));
	else
		tmp = (m * (1.0 - m)) / (v / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-28], N[(N[(1.0 + m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.0000000000000002e-28

   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 0 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      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 100.0%

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

      8. sqr-neg 100.0%

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

      9. sqrt-unprod 100.0%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   7. Simplified 100.0%

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

   if 5.0000000000000002e-28 < 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. 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. Taylor expanded in v around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.8%

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

      3. mul-1-neg 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. unpow2 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. distribute-lft-in 99.8%

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

      8. sub-neg 99.8%

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

      9. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-28}:\\ \;\;\;\;\left(1 + m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{1 - m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = (m * (m / v)) * (m + -1.0)
	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(m / v)) * Float64(m + -1.0));
	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 * (m / v)) * (m + -1.0);
	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[(m / v), $MachinePrecision]), $MachinePrecision] * N[(m + -1.0), $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. Add Preprocessing

   3. Taylor expanded in m around 0 97.6%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. 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. Taylor expanded in v around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.9%

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

      3. mul-1-neg 99.9%

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

      4. *-rgt-identity 99.9%

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

      5. unpow2 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. distribute-lft-in 99.9%

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

      8. sub-neg 99.9%

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

      9. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf 98.2%

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

      1. neg-mul-1 98.2%

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

      2. distribute-neg-frac2 98.2%

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

   10. Simplified 98.2%

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

4. Final simplification 97.9%

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

Alternative 7: 98.1% accurate, 0.9× speedup

Math

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

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(m / Float64(Float64(v / Float64(m + -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[(m / N[(N[(v / N[(m + -1.0), $MachinePrecision]), $MachinePrecision] / 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. Add Preprocessing

   3. Taylor expanded in m around 0 97.6%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Add Preprocessing
   3. 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. Taylor expanded in v around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.9%

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

      3. mul-1-neg 99.9%

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

      4. *-rgt-identity 99.9%

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

      5. unpow2 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. distribute-lft-in 99.9%

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

      8. sub-neg 99.9%

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

      9. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf 98.2%

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

      1. neg-mul-1 98.2%

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

      2. distribute-neg-frac2 98.2%

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

   10. Simplified 98.2%

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

       1. *-commutative 98.2%

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

       2. *-commutative 98.2%

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

       3. associate-*l* 98.2%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 0.1%

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

       6. sqr-neg 0.1%

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

       7. sqrt-unprod 0.1%

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

       8. add-sqr-sqrt 0.1%

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

       9. associate-/r/ 0.1%

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

       10. frac-2neg 0.1%

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

       11. distribute-neg-frac 0.1%

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

       12. add-sqr-sqrt 0.0%

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

       13. sqrt-unprod 96.6%

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

       14. sqr-neg 96.6%

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

       15. sqrt-unprod 98.1%

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

       16. add-sqr-sqrt 98.2%

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

       17. *-commutative 98.2%

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

       18. associate-/r* 98.2%

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

   12. Applied egg-rr 98.2%

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

4. Final simplification 97.9%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m * N[(N[(1.0 - m), $MachinePrecision] / 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{-177}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 + m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5.6e-177) -1.0 (/ (* m (+ 1.0 m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.6e-177) {
		tmp = -1.0;
	} else {
		tmp = (m * (1.0 + 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 <= 5.6d-177) then
        tmp = -1.0d0
    else
        tmp = (m * (1.0d0 + m)) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5.6e-177) {
		tmp = -1.0;
	} else {
		tmp = (m * (1.0 + m)) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.6e-177:
		tmp = -1.0
	else:
		tmp = (m * (1.0 + m)) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.6e-177)
		tmp = -1.0;
	else
		tmp = Float64(Float64(m * Float64(1.0 + m)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.6e-177)
		tmp = -1.0;
	else
		tmp = (m * (1.0 + m)) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.6e-177], -1.0, N[(N[(m * N[(1.0 + m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.59999999999999973e-177

   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 75.2%

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

   if 5.59999999999999973e-177 < 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 34.1%

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

      1. sub-neg 34.1%

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

      2. distribute-rgt-in 34.1%

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

      3. *-un-lft-identity 34.1%

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

      4. sub-neg 34.1%

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

      5. metadata-eval 34.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 79.0%

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

      8. sqr-neg 79.0%

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

      9. sqrt-unprod 79.0%

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

      10. add-sqr-sqrt 79.0%

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

      11. sub-neg 79.0%

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

      12. metadata-eval 79.0%

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

   5. Applied egg-rr 79.0%

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

      1. distribute-rgt1-in 79.0%

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

   7. Simplified 79.0%

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

   8. Taylor expanded in v around 0 70.4%

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

      1. +-commutative 70.4%

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

   10. Simplified 70.4%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{-177}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 + m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 + m), $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 47.0%

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

   1. sub-neg 47.0%

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

   2. distribute-rgt-in 47.0%

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

   3. *-un-lft-identity 47.0%

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

   4. sub-neg 47.0%

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

   5. metadata-eval 47.0%

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

   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 83.1%

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

   8. sqr-neg 83.1%

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

   9. sqrt-unprod 83.1%

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

   10. add-sqr-sqrt 83.1%

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

   11. sub-neg 83.1%

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

   12. metadata-eval 83.1%

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

5. Applied egg-rr 83.1%

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

   1. distribute-rgt1-in 83.1%

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

7. Simplified 83.1%

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

8. Final simplification 83.1%

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

Alternative 11: 62.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 7.2e-173) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.2e-173) {
		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 <= 7.2d-173) 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 <= 7.2e-173) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.2e-173:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.2e-173)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7.2e-173)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.2e-173], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.19999999999999943e-173

   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 75.2%

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

   if 7.19999999999999943e-173 < 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. 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. Taylor expanded in v around 0 91.2%

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

      1. *-commutative 91.2%

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

      2. associate-/l* 91.3%

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

      3. mul-1-neg 91.3%

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

      4. *-rgt-identity 91.3%

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

      5. unpow2 91.3%

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

      6. distribute-rgt-neg-in 91.3%

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

      7. distribute-lft-in 91.3%

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

      8. sub-neg 91.3%

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

      9. associate-/l* 91.2%

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

   7. Simplified 91.2%

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

   8. Taylor expanded in m around 0 25.5%

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

   9. Taylor expanded in m around 0 54.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.7% 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 23.7%

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

   1. neg-mul-1 23.7%

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

   2. neg-sub0 23.7%

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

   3. associate--r- 23.7%

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

   4. metadata-eval 23.7%

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

7. Simplified 23.7%

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

8. Final simplification 23.7%

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

Alternative 13: 24.3% 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 21.2%

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

6. Final simplification 21.2%

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

Reproduce

herbie shell --seed 2024039 
(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)))