b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 5.9s

Alternatives: 12

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

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

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 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 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 4: 98.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.4d0) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = (m + (-2.0d0)) * (m * (m / v))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. flip-- 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 96.9%

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

      1. sub-neg 96.9%

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

      2. *-commutative 96.9%

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

      3. distribute-lft-in 96.9%

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

      4. *-rgt-identity 96.9%

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

      5. associate-*r/ 97.1%

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

      6. *-rgt-identity 97.1%

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

      7. metadata-eval 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in v around 0 97.1%

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

   if 2.39999999999999991 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. flip-- 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 23.6%

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

      1. unpow2 23.6%

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

      2. cube-mult 23.6%

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

      3. associate-*r/ 23.6%

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

      4. distribute-rgt-out 99.7%

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

      5. associate-*r/ 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 98.3%

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

Alternative 5: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- 1.0 m) (+ -1.0 (/ m v))) (* (+ m -2.0) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (m / v))
    else
        tmp = (m + (-2.0d0)) * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = (m + -2.0) * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(m + -2.0) * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	else
		tmp = (m + -2.0) * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m + -2.0), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 97.2%

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

   if 1.6000000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. flip-- 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 23.6%

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

      1. unpow2 23.6%

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

      2. cube-mult 23.6%

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

      3. associate-*r/ 23.6%

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

      4. distribute-rgt-out 99.7%

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

      5. associate-*r/ 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 6: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} \cdot \left(m + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- 1.0 m) (+ -1.0 (/ m v))) (* (/ m (/ v m)) (+ m -2.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = (m / (v / m)) * (m + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (m / v))
    else
        tmp = (m / (v / m)) * (m + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = (m / (v / m)) * (m + -2.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = (m / (v / m)) * (m + -2.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(m / Float64(v / m)) * Float64(m + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	else
		tmp = (m / (v / m)) * (m + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 97.2%

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

   if 1.6000000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. flip-- 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around inf 23.6%

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

      1. unpow2 23.6%

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

      2. cube-mult 23.6%

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

      3. associate-*r/ 23.6%

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

      4. distribute-rgt-out 99.7%

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

      5. associate-*r/ 99.7%

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

   8. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} \cdot \left(m + -2\right)\\ \end{array} \]

Alternative 7: 87.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(-1.0 + 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 <= 2.4)
		tmp = -1.0 + (m / v);
	else
		tmp = (m / v) * (m + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. flip-- 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 96.9%

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

      1. sub-neg 96.9%

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

      2. *-commutative 96.9%

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

      3. distribute-lft-in 96.9%

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

      4. *-rgt-identity 96.9%

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

      5. associate-*r/ 97.1%

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

      6. *-rgt-identity 97.1%

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

      7. metadata-eval 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in v around 0 97.1%

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

   if 2.39999999999999991 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 0.1%

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

   3. Taylor expanded in v around 0 0.1%

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

      1. associate-/l* 0.1%

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

      2. associate-/r/ 0.1%

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

      3. sub-neg 0.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 78.1%

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

      6. sqr-neg 78.1%

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

      7. sqrt-unprod 78.1%

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

      8. add-sqr-sqrt 78.1%

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

      9. +-commutative 78.1%

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

   5. Applied egg-rr 78.1%

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

4. Final simplification 88.4%

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

Alternative 8: 87.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. flip-- 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in m around 0 96.9%

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

      1. sub-neg 96.9%

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

      2. *-commutative 96.9%

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

      3. distribute-lft-in 96.9%

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

      4. *-rgt-identity 96.9%

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

      5. associate-*r/ 97.1%

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

      6. *-rgt-identity 97.1%

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

      7. metadata-eval 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in v around 0 97.1%

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

   if 2.39999999999999991 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 0.1%

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

   3. Taylor expanded in v around 0 0.1%

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

      1. associate-/l* 0.1%

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

   5. Simplified 0.1%

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

      1. clear-num 0.1%

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

      2. associate-/r/ 0.1%

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

   7. Applied egg-rr 0.1%

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

      1. add-log-exp 87.1%

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

      2. *-un-lft-identity 87.1%

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

      3. log-prod 87.1%

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

      4. metadata-eval 87.1%

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

      5. add-log-exp 43.7%

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

      6. associate-*l/ 43.7%

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

      7. *-un-lft-identity 43.7%

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

      8. *-un-lft-identity 43.7%

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

      9. *-un-lft-identity 43.7%

         \[\leadsto \frac{m}{0 + \frac{v}{\color{blue}{1 - m}}} \]

      10. sub-neg 43.7%

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

      11. +-commutative 43.7%

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

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{m}{0 + \frac{v}{\color{blue}{\sqrt{-m} \cdot \sqrt{-m}} + 1}} \]

      13. sqrt-unprod 78.1%

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

      14. sqr-neg 78.1%

          \[\leadsto \frac{m}{0 + \frac{v}{\sqrt{\color{blue}{m \cdot m}} + 1}} \]

      15. sqrt-unprod 78.1%

          \[\leadsto \frac{m}{0 + \frac{v}{\color{blue}{\sqrt{m} \cdot \sqrt{m}} + 1}} \]

      16. add-sqr-sqrt 78.1%

          \[\leadsto \frac{m}{0 + \frac{v}{\color{blue}{m} + 1}} \]

   9. Applied egg-rr 78.1%

      \[\leadsto \frac{m}{\color{blue}{0 + \frac{v}{m + 1}}} \]
   10. Step-by-step derivation

       1. +-lft-identity 78.1%

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

   11. Simplified 78.1%

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

4. Final simplification 88.4%

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

Alternative 9: 62.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-158}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.6e-158) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-158) {
		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 <= 1.6d-158) 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 <= 1.6e-158) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6e-158:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6e-158)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6e-158)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6e-158], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.59999999999999998e-158

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 75.1%

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

   if 1.59999999999999998e-158 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 37.1%

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

   3. Taylor expanded in v around 0 27.0%

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

      1. associate-/l* 27.0%

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

   5. Simplified 27.0%

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

   6. Taylor expanded in m around 0 60.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-158}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 10: 76.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. flip-- 99.9%

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

   2. associate-*r/ 99.9%

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

   3. metadata-eval 99.9%

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

   4. +-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in m around 0 77.8%

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

   1. sub-neg 77.8%

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

   2. *-commutative 77.8%

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

   3. distribute-lft-in 77.8%

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

   4. *-rgt-identity 77.8%

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

   5. associate-*r/ 77.9%

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

   6. *-rgt-identity 77.9%

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

   7. metadata-eval 77.9%

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

8. Simplified 77.9%

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

9. Taylor expanded in v around 0 77.9%

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

10. Final simplification 77.9%

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

Alternative 11: 27.0% 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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around inf 27.2%

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

   1. neg-mul-1 27.2%

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

   2. neg-sub0 27.2%

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

   3. associate--r- 27.2%

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

   4. metadata-eval 27.2%

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

6. Simplified 27.2%

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

7. Final simplification 27.2%

   \[\leadsto m + -1 \]

Alternative 12: 24.6% 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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 25.0%

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

5. Final simplification 25.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023257 
(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)))