b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.2s

Alternatives: 16

Speedup: 1.0×

• 41.8% 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 100.0%

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

   1. sub-neg 100.0%

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

   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%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 95.9%

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

   3. Taylor expanded in v around 0 95.9%

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

      1. +-commutative 95.9%

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

      2. mul-1-neg 95.9%

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

      3. unsub-neg 95.9%

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

   5. Simplified 95.9%

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

   if 1 < m

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. associate-*l/ 99.1%

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

      4. distribute-rgt-neg-out 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in v around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. *-commutative 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 97.6%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m - m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.3e-25) (+ m (+ -1.0 (/ m v))) (/ (- m (* m m)) (/ v (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.3e-25) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (m - (m * m)) / (v / (1.0 - m));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.3d-25) then
        tmp = m + ((-1.0d0) + (m / v))
    else
        tmp = (m - (m * m)) / (v / (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.3e-25) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (m - (m * m)) / (v / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.3e-25:
		tmp = m + (-1.0 + (m / v))
	else:
		tmp = (m - (m * m)) / (v / (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.3e-25)
		tmp = Float64(m + Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(m - Float64(m * m)) / Float64(v / Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.3e-25)
		tmp = m + (-1.0 + (m / v));
	else
		tmp = (m - (m * m)) / (v / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.3e-25], N[(m + N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.3e-25

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. *-lft-identity 99.8%

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

      3. associate--l+ 99.8%

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

      4. associate-*l/ 100.0%

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

      5. *-lft-identity 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if 1.3e-25 < 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. 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%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. mul-1-neg 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m - m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 4: 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 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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

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 100.0%

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

2. Final simplification 100.0%

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

Alternative 6: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 95.6%

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

      1. distribute-rgt-in 95.6%

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

      2. *-lft-identity 95.6%

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

      3. associate--l+ 95.6%

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

      4. associate-*l/ 95.8%

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

      5. *-lft-identity 95.8%

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

      6. sub-neg 95.8%

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

      7. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   if 1 < m

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. associate-*l/ 99.1%

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

      4. distribute-rgt-neg-out 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in m around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. unpow2 99.1%

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

      4. metadata-eval 99.1%

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

      5. distribute-lft1-in 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-commutative 99.1%

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

      8. neg-mul-1 99.1%

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

      9. sub-neg 99.1%

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

      10. associate-/l* 99.1%

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

      11. associate-/r/ 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in m around inf 28.7%

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

       1. mul-1-neg 28.7%

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

       2. unpow2 28.7%

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

       3. associate-/l* 28.7%

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

       4. neg-sub0 28.7%

          \[\leadsto \color{blue}{\left(0 - \frac{m}{\frac{v}{m}}\right)} + \frac{{m}^{3}}{v} \]

       5. associate-+l- 28.7%

          \[\leadsto \color{blue}{0 - \left(\frac{m}{\frac{v}{m}} - \frac{{m}^{3}}{v}\right)} \]

       6. cube-mult 28.7%

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

       7. associate-*l/ 28.7%

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

       8. cancel-sign-sub-inv 28.7%

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

       9. *-rgt-identity 28.7%

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

       10. distribute-lft-neg-in 28.7%

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

       11. associate-*r* 28.7%

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

       12. associate-/r/ 28.7%

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

       13. distribute-rgt-neg-out 28.7%

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

       14. distribute-lft-in 99.1%

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

       15. sub-neg 99.1%

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

       16. *-commutative 99.1%

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

       17. neg-sub0 99.1%

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

       18. distribute-lft-neg-in 99.1%

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

       19. associate-*r/ 99.0%

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

   12. Simplified 99.0%

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

       1. div-inv 99.0%

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

       2. *-commutative 99.0%

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

       3. clear-num 99.0%

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

       4. associate-*l* 99.0%

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

       5. +-commutative 99.0%

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

   14. Applied egg-rr 99.0%

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

4. Final simplification 97.5%

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

Alternative 7: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 95.6%

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

      1. distribute-rgt-in 95.6%

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

      2. *-lft-identity 95.6%

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

      3. associate--l+ 95.6%

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

      4. associate-*l/ 95.8%

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

      5. *-lft-identity 95.8%

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

      6. sub-neg 95.8%

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

      7. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   if 1 < m

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. associate-*l/ 99.1%

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

      4. distribute-rgt-neg-out 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in m around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. unpow2 99.1%

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

      4. metadata-eval 99.1%

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

      5. distribute-lft1-in 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-commutative 99.1%

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

      8. neg-mul-1 99.1%

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

      9. sub-neg 99.1%

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

      10. associate-/l* 99.1%

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

      11. associate-/r/ 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in m around inf 28.7%

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

       1. mul-1-neg 28.7%

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

       2. unpow2 28.7%

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

       3. associate-/l* 28.7%

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

       4. neg-sub0 28.7%

          \[\leadsto \color{blue}{\left(0 - \frac{m}{\frac{v}{m}}\right)} + \frac{{m}^{3}}{v} \]

       5. associate-+l- 28.7%

          \[\leadsto \color{blue}{0 - \left(\frac{m}{\frac{v}{m}} - \frac{{m}^{3}}{v}\right)} \]

       6. cube-mult 28.7%

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

       7. associate-*l/ 28.7%

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

       8. cancel-sign-sub-inv 28.7%

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

       9. *-rgt-identity 28.7%

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

       10. distribute-lft-neg-in 28.7%

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

       11. associate-*r* 28.7%

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

       12. associate-/r/ 28.7%

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

       13. distribute-rgt-neg-out 28.7%

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

       14. distribute-lft-in 99.1%

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

       15. sub-neg 99.1%

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

       16. *-commutative 99.1%

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

       17. neg-sub0 99.1%

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

       18. distribute-lft-neg-in 99.1%

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

       19. associate-*r/ 99.0%

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

   12. Simplified 99.0%

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

       1. div-inv 99.0%

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

       2. clear-num 99.0%

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

       3. associate-*l* 99.1%

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

       4. +-commutative 99.1%

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

       5. associate-*r/ 99.1%

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

   14. Applied egg-rr 99.1%

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

4. Final simplification 97.5%

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

Alternative 8: 97.9% accurate, 1.2× speedup

Math

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

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(Float64(m * 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[(N[(m * m), $MachinePrecision] / v), $MachinePrecision] * N[(m + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 95.9%

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

   if 1 < m

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. associate-*l/ 99.1%

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

      4. distribute-rgt-neg-out 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in m around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. unpow2 99.1%

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

      4. metadata-eval 99.1%

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

      5. distribute-lft1-in 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-commutative 99.1%

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

      8. neg-mul-1 99.1%

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

      9. sub-neg 99.1%

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

      10. associate-/l* 99.1%

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

      11. associate-/r/ 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in m around inf 28.7%

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

       1. mul-1-neg 28.7%

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

       2. unpow2 28.7%

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

       3. associate-/l* 28.7%

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

       4. neg-sub0 28.7%

          \[\leadsto \color{blue}{\left(0 - \frac{m}{\frac{v}{m}}\right)} + \frac{{m}^{3}}{v} \]

       5. associate-+l- 28.7%

          \[\leadsto \color{blue}{0 - \left(\frac{m}{\frac{v}{m}} - \frac{{m}^{3}}{v}\right)} \]

       6. cube-mult 28.7%

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

       7. associate-*l/ 28.7%

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

       8. cancel-sign-sub-inv 28.7%

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

       9. *-rgt-identity 28.7%

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

       10. distribute-lft-neg-in 28.7%

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

       11. associate-*r* 28.7%

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

       12. associate-/r/ 28.7%

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

       13. distribute-rgt-neg-out 28.7%

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

       14. distribute-lft-in 99.1%

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

       15. sub-neg 99.1%

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

       16. *-commutative 99.1%

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

       17. neg-sub0 99.1%

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

       18. distribute-lft-neg-in 99.1%

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

       19. associate-*r/ 99.0%

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

   12. Simplified 99.0%

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

       1. div-inv 99.0%

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

       2. clear-num 99.0%

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

       3. associate-*l* 99.1%

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

       4. +-commutative 99.1%

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

       5. associate-*r/ 99.1%

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

   14. Applied egg-rr 99.1%

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

4. Final simplification 97.6%

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

Alternative 9: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

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) * (m + -1.0)) / v;
	}
	return tmp;
}

Fortran

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 95.9%

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

   if 1 < m

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. associate-*l/ 99.1%

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

      4. distribute-rgt-neg-out 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in v around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. *-commutative 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 97.6%

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

Alternative 10: 97.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000009

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 95.6%

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

      1. distribute-rgt-in 95.6%

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

      2. *-lft-identity 95.6%

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

      3. associate--l+ 95.6%

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

      4. associate-*l/ 95.8%

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

      5. *-lft-identity 95.8%

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

      6. sub-neg 95.8%

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

      7. metadata-eval 95.8%

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

   6. Simplified 95.8%

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

   if 2.60000000000000009 < m

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. unpow2 99.1%

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

      3. associate-*l/ 99.1%

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

      4. distribute-rgt-neg-out 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in m around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. *-commutative 99.1%

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

      3. unpow2 99.1%

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

      4. metadata-eval 99.1%

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

      5. distribute-lft1-in 99.1%

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

      6. distribute-rgt1-in 99.1%

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

      7. *-commutative 99.1%

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

      8. neg-mul-1 99.1%

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

      9. sub-neg 99.1%

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

      10. associate-/l* 99.1%

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

      11. associate-/r/ 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in m around inf 28.7%

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

       1. mul-1-neg 28.7%

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

       2. unpow2 28.7%

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

       3. associate-/l* 28.7%

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

       4. neg-sub0 28.7%

          \[\leadsto \color{blue}{\left(0 - \frac{m}{\frac{v}{m}}\right)} + \frac{{m}^{3}}{v} \]

       5. associate-+l- 28.7%

          \[\leadsto \color{blue}{0 - \left(\frac{m}{\frac{v}{m}} - \frac{{m}^{3}}{v}\right)} \]

       6. cube-mult 28.7%

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

       7. associate-*l/ 28.7%

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

       8. cancel-sign-sub-inv 28.7%

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

       9. *-rgt-identity 28.7%

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

       10. distribute-lft-neg-in 28.7%

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

       11. associate-*r* 28.7%

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

       12. associate-/r/ 28.7%

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

       13. distribute-rgt-neg-out 28.7%

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

       14. distribute-lft-in 99.1%

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

       15. sub-neg 99.1%

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

       16. *-commutative 99.1%

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

       17. neg-sub0 99.1%

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

       18. distribute-lft-neg-in 99.1%

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

       19. associate-*r/ 99.0%

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

   12. Simplified 99.0%

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

   13. Taylor expanded in m around inf 98.9%

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

       1. unpow2 98.9%

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

   15. Simplified 98.9%

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

4. Final simplification 97.4%

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

Alternative 11: 63.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.85 \cdot 10^{-155}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.85e-155) -1.0 (+ m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.85e-155) {
		tmp = -1.0;
	} 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 <= 1.85d-155) then
        tmp = -1.0d0
    else
        tmp = m + (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.85e-155) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.85e-155:
		tmp = -1.0
	else:
		tmp = m + (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.85e-155)
		tmp = -1.0;
	else
		tmp = Float64(m + Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.85e-155)
		tmp = -1.0;
	else
		tmp = m + (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.85e-155], -1.0, N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.85e-155

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 80.6%

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

   if 1.85e-155 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 63.0%

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

      1. distribute-rgt-in 63.0%

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

      2. *-lft-identity 63.0%

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

      3. associate--l+ 63.0%

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

      4. associate-*l/ 63.1%

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

      5. *-lft-identity 63.1%

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

      6. sub-neg 63.1%

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

      7. metadata-eval 63.1%

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

   6. Simplified 63.1%

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

   7. Taylor expanded in m around inf 55.5%

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

      1. distribute-rgt-in 55.5%

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

      2. *-lft-identity 55.5%

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

      3. associate-*l/ 55.6%

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

      4. *-lft-identity 55.6%

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

   9. Simplified 55.6%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.85 \cdot 10^{-155}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \]

Alternative 12: 76.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 71.1%

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

   1. distribute-rgt-in 71.1%

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

   2. *-lft-identity 71.1%

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

   3. associate--l+ 71.1%

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

   4. associate-*l/ 71.2%

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

   5. *-lft-identity 71.2%

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

   6. sub-neg 71.2%

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

   7. metadata-eval 71.2%

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

6. Simplified 71.2%

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

7. Final simplification 71.2%

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

Alternative 13: 63.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.05 \cdot 10^{-155}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.05e-155) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.05e-155) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.05d-155) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.05e-155) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.05e-155:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.05e-155)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.05e-155)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.05e-155], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.0499999999999999e-155

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 80.6%

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

   if 2.0499999999999999e-155 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 63.0%

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

      1. distribute-rgt-in 63.0%

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

      2. *-lft-identity 63.0%

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

      3. associate--l+ 63.0%

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

      4. associate-*l/ 63.1%

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

      5. *-lft-identity 63.1%

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

      6. sub-neg 63.1%

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

      7. metadata-eval 63.1%

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

   6. Simplified 63.1%

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

   7. Taylor expanded in m around inf 55.5%

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

      1. distribute-rgt-in 55.5%

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

      2. *-lft-identity 55.5%

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

      3. associate-*l/ 55.6%

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

      4. *-lft-identity 55.6%

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

   9. Simplified 55.6%

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

   10. Taylor expanded in v around 0 55.6%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.05 \cdot 10^{-155}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 14: 27.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.8 \cdot 10^{-26}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 7.8e-26) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.8e-26) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 7.8d-26) then
        tmp = -1.0d0
    else
        tmp = m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7.8e-26) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.8e-26:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.8e-26)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7.8e-26)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.8e-26], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 7.79999999999999973e-26

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 54.6%

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

   if 7.79999999999999973e-26 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 50.2%

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

      1. distribute-rgt-in 50.2%

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

      2. *-lft-identity 50.2%

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

      3. associate--l+ 50.2%

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

      4. associate-*l/ 50.2%

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

      5. *-lft-identity 50.2%

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

      6. sub-neg 50.2%

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

      7. metadata-eval 50.2%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in m around inf 50.2%

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

      1. distribute-rgt-in 50.2%

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

      2. *-lft-identity 50.2%

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

      3. associate-*l/ 50.2%

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

      4. *-lft-identity 50.2%

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

   9. Simplified 50.2%

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

   10. Taylor expanded in v around inf 5.4%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.8 \cdot 10^{-26}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]

Alternative 15: 27.6% 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 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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around inf 25.9%

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

   1. neg-mul-1 25.9%

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

   2. neg-sub0 25.9%

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

   3. associate--r- 25.9%

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

   4. metadata-eval 25.9%

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

6. Simplified 25.9%

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

7. Final simplification 25.9%

   \[\leadsto m + -1 \]

Alternative 16: 25.1% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 23.5%

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

5. Final simplification 23.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023293 
(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)))