b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.6s

Alternatives: 14

Speedup: 1.0×

• 42.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

   3. Taylor expanded in m around 0 97.2%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 98.5%

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

      1. neg-mul-1 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 97.9%

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

Alternative 3: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

   3. Taylor expanded in m around 0 97.2%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r/ 99.9%

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

      4. clear-num 99.8%

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

      5. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around inf 98.5%

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

      1. neg-mul-1 98.5%

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

   9. Simplified 98.5%

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

4. Final simplification 97.8%

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

Alternative 4: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 97.2%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around inf 98.5%

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

      1. neg-mul-1 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 97.8%

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

Alternative 5: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   3. Taylor expanded in m around 0 97.2%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around inf 98.5%

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

      1. neg-mul-1 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in m around inf 98.4%

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

      1. neg-mul-1 98.5%

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

   10. Simplified 98.4%

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

4. Final simplification 97.8%

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

Alternative 6: 87.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = 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(Float64(m / v) + -1.0));
	else
		tmp = Float64(m * Float64(Float64(m + 1.0) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = 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[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m + 1.0), $MachinePrecision] / v), $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. Add Preprocessing

   3. Taylor expanded in m around 0 97.2%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 0.1%

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

   6. Taylor expanded in v around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. rem-square-sqrt 0.0%

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

      3. associate-*r* 0.0%

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

      4. /-rgt-identity 0.0%

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

      5. associate-/r/ 0.0%

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

      6. associate-*l/ 0.0%

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

      7. times-frac 0.0%

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

      8. rem-square-sqrt 0.1%

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

      9. associate-*r/ 0.1%

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

      10. *-rgt-identity 0.1%

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

   8. Simplified 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. mul-1-neg 0.1%

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

      5. mul-1-neg 0.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 72.4%

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

      8. mul-1-neg 72.4%

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

      9. distribute-neg-frac 72.4%

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

      10. neg-sub0 30.6%

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

   10. Applied egg-rr 30.6%

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

       1. neg-sub0 72.4%

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

       2. distribute-neg-frac 72.4%

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

   12. Simplified 72.4%

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

       1. associate-/r/ 72.4%

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

       2. sub-neg 72.4%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 0.1%

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

       5. sqr-neg 0.1%

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

       6. sqrt-unprod 0.1%

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

       7. add-sqr-sqrt 0.1%

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

       8. add-sqr-sqrt 0.0%

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

       9. sqrt-unprod 77.4%

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

       10. sqr-neg 77.4%

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

       11. sqrt-unprod 72.4%

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

       12. add-sqr-sqrt 72.4%

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

   14. Applied egg-rr 72.4%

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

4. Final simplification 84.7%

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

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*r/ 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

Alternative 9: 87.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.3:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.3) (+ -1.0 (+ m (/ m v))) (* m (/ (+ m 1.0) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.3) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.3d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = m * ((m + 1.0d0) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.3) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.3:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = m * ((m + 1.0) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.3)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(m * Float64(Float64(m + 1.0) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.3)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = m * ((m + 1.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.3], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m + 1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.2999999999999998

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 96.9%

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

      1. fma-neg 96.9%

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

      2. metadata-eval 96.9%

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

   7. Simplified 96.9%

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

   8. Taylor expanded in v around inf 97.1%

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

   if 2.2999999999999998 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 0.1%

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

   6. Taylor expanded in v around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. rem-square-sqrt 0.0%

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

      3. associate-*r* 0.0%

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

      4. /-rgt-identity 0.0%

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

      5. associate-/r/ 0.0%

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

      6. associate-*l/ 0.0%

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

      7. times-frac 0.0%

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

      8. rem-square-sqrt 0.1%

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

      9. associate-*r/ 0.1%

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

      10. *-rgt-identity 0.1%

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

   8. Simplified 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. mul-1-neg 0.1%

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

      5. mul-1-neg 0.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 72.4%

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

      8. mul-1-neg 72.4%

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

      9. distribute-neg-frac 72.4%

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

      10. neg-sub0 30.6%

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

   10. Applied egg-rr 30.6%

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

       1. neg-sub0 72.4%

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

       2. distribute-neg-frac 72.4%

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

   12. Simplified 72.4%

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

       1. associate-/r/ 72.4%

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

       2. sub-neg 72.4%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 0.1%

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

       5. sqr-neg 0.1%

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

       6. sqrt-unprod 0.1%

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

       7. add-sqr-sqrt 0.1%

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

       8. add-sqr-sqrt 0.0%

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

       9. sqrt-unprod 77.4%

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

       10. sqr-neg 77.4%

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

       11. sqrt-unprod 72.4%

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

       12. add-sqr-sqrt 72.4%

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

   14. Applied egg-rr 72.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.3:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.5 \cdot 10^{-119}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 9.5e-119) -1.0 (* (/ m v) (+ m 1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 9.5e-119) {
		tmp = -1.0;
	} 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 <= 9.5d-119) then
        tmp = -1.0d0
    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 <= 9.5e-119) {
		tmp = -1.0;
	} else {
		tmp = (m / v) * (m + 1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 9.5e-119:
		tmp = -1.0
	else:
		tmp = (m / v) * (m + 1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 9.5e-119)
		tmp = -1.0;
	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 <= 9.5e-119)
		tmp = -1.0;
	else
		tmp = (m / v) * (m + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 9.5e-119], -1.0, N[(N[(m / v), $MachinePrecision] * N[(m + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.5000000000000002e-119

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 73.0%

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

   if 9.5000000000000002e-119 < 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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 25.5%

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

   6. Taylor expanded in v around 0 22.2%

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

      1. *-commutative 22.2%

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

      2. rem-square-sqrt 22.2%

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

      3. associate-*r* 22.2%

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

      4. /-rgt-identity 22.2%

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

      5. associate-/r/ 22.1%

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

      6. associate-*l/ 22.0%

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

      7. times-frac 22.0%

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

      8. rem-square-sqrt 22.1%

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

      9. associate-*r/ 22.2%

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

      10. *-rgt-identity 22.2%

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

   8. Simplified 22.2%

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

      1. div-inv 22.2%

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

      2. clear-num 22.2%

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

      3. un-div-inv 22.1%

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

      4. *-commutative 22.1%

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

      5. sub-neg 22.1%

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

      6. distribute-lft-in 22.1%

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

      7. *-commutative 22.1%

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

      8. *-un-lft-identity 22.1%

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

      9. un-div-inv 22.1%

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

      10. add-sqr-sqrt 22.1%

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

      11. sqrt-unprod 10.2%

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

      12. sqr-neg 10.2%

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

      13. mul-1-neg 10.2%

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

      14. mul-1-neg 10.2%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 74.5%

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

      17. associate-/r/ 74.5%

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

      18. mul-1-neg 74.5%

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

      19. frac-2neg 74.5%

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

      20. un-div-inv 74.6%

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

      21. div-inv 74.6%

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

      22. clear-num 74.6%

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

   10. Applied egg-rr 74.6%

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

       1. *-lft-identity 74.6%

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

       2. distribute-rgt-in 74.6%

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

       3. +-commutative 74.6%

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

   12. Simplified 74.6%

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

Alternative 11: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-119}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 7e-119) -1.0 (/ 1.0 (/ v m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7e-119) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / (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 <= 7d-119) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 / (v / m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7e-119) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / (v / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7e-119:
		tmp = -1.0
	else:
		tmp = 1.0 / (v / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7e-119)
		tmp = -1.0;
	else
		tmp = Float64(1.0 / Float64(v / m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7e-119)
		tmp = -1.0;
	else
		tmp = 1.0 / (v / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7e-119], -1.0, N[(1.0 / N[(v / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7e-119

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 73.0%

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

   if 7e-119 < 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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 25.5%

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

   6. Taylor expanded in v around 0 22.2%

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

      1. *-commutative 22.2%

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

      2. rem-square-sqrt 22.2%

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

      3. associate-*r* 22.2%

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

      4. /-rgt-identity 22.2%

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

      5. associate-/r/ 22.1%

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

      6. associate-*l/ 22.0%

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

      7. times-frac 22.0%

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

      8. rem-square-sqrt 22.1%

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

      9. associate-*r/ 22.2%

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

      10. *-rgt-identity 22.2%

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

   8. Simplified 22.2%

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

   9. Taylor expanded in m around 0 57.5%

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

Alternative 12: 25.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{-61}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.3e-61) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.3e-61) {
		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 <= 1.3d-61) 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 <= 1.3e-61) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.3e-61:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.3e-61)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.3e-61)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.3e-61], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 1.30000000000000005e-61

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 60.9%

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

   if 1.30000000000000005e-61 < 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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 55.7%

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

      1. fma-neg 55.7%

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

      2. metadata-eval 55.7%

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

   7. Simplified 55.7%

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

   8. Taylor expanded in v around inf 4.6%

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

   9. Taylor expanded in m around inf 5.2%

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

Alternative 13: 26.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 26.4%

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

   1. neg-mul-1 26.4%

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

   2. neg-sub0 26.4%

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

   3. associate--r- 26.4%

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

   4. metadata-eval 26.4%

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

7. Simplified 26.4%

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

8. Final simplification 26.4%

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

Alternative 14: 23.7% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 24.0%

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

Reproduce

herbie shell --seed 2024146 
(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)))