b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

Alternatives: 14

Speedup: 1.0×

• 41.6% 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0)
   (* (- 1.0 m) (+ -1.0 (/ 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) * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) * (-1.0 - (m * (m / v)));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} 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) * (-1.0 + (m / v))
	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(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 - Float64(m * Float64(m / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	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[(-1.0 + N[(m / v), $MachinePrecision]), $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 100.0%

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

   3. Taylor expanded in m around 0 98.8%

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

   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 98.6%

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

      1. neg-mul-1 98.6%

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

      2. distribute-neg-frac2 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 98.7%

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

Alternative 3: 97.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0)
   (* (- 1.0 m) (+ -1.0 (/ 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) * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) * (-1.0 - ((m * m) / v));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} 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) * (-1.0 + (m / v))
	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(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 - Float64(Float64(m * m) / v)));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 98.8%

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

   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 98.6%

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

      1. neg-mul-1 98.6%

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

      2. distribute-neg-frac2 98.6%

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

   7. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. distribute-frac-neg2 98.6%

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

      3. distribute-frac-neg 98.6%

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

      4. associate-*l/ 98.6%

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

   9. Applied egg-rr 98.6%

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

4. Final simplification 98.7%

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

Alternative 4: 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. Final simplification 99.8%

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

Alternative 5: 72.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.5 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.5e-184) -1.0 (* m (/ (+ m 1.0) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.5e-184) {
		tmp = -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 <= 2.5d-184) then
        tmp = -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 <= 2.5e-184) {
		tmp = -1.0;
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.5e-184:
		tmp = -1.0
	else:
		tmp = m * ((m + 1.0) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.5e-184)
		tmp = -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 <= 2.5e-184)
		tmp = -1.0;
	else
		tmp = m * ((m + 1.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.5e-184], -1.0, N[(m * N[(N[(m + 1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.50000000000000001e-184

   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 74.4%

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

   if 2.50000000000000001e-184 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 37.1%

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

   4. Taylor expanded in v around 0 37.1%

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

      1. +-commutative 37.1%

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

      2. associate-*r* 37.1%

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

      3. distribute-rgt-out 37.1%

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

      4. mul-1-neg 37.1%

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

      5. unsub-neg 37.1%

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

   6. Simplified 37.1%

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

      1. div-inv 37.0%

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

      2. sub-neg 37.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 85.1%

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

      5. sqr-neg 85.1%

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

      6. sqrt-unprod 85.1%

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

      7. add-sqr-sqrt 85.1%

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

   8. Applied egg-rr 85.1%

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

   9. Taylor expanded in v around 0 77.1%

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

       1. associate-/l* 77.0%

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

       2. +-commutative 77.0%

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

   11. Simplified 77.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.5 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.3) {
		tmp = (v - m) * (-1.0 / 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 = (v - m) * ((-1.0d0) / 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 = (v - m) * (-1.0 / v);
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.3)
		tmp = Float64(Float64(v - m) * Float64(-1.0 / 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 = (v - m) * (-1.0 / v);
	else
		tmp = m * ((m + 1.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.3], N[(N[(v - m), $MachinePrecision] * N[(-1.0 / v), $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 100.0%

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

   3. Taylor expanded in m around 0 98.8%

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

   4. Taylor expanded in v around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-*r* 98.7%

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

      3. distribute-rgt-out 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in v around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-*r* 98.7%

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

      3. distribute-rgt-out 98.7%

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

      4. sub-neg 98.7%

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

      5. metadata-eval 98.7%

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

      6. distribute-neg-in 98.7%

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

      7. neg-mul-1 98.7%

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

      8. sub-neg 98.7%

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

      9. distribute-lft-neg-in 98.7%

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

      10. *-commutative 98.7%

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

      11. distribute-neg-frac 98.7%

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

      12. associate-/l* 98.4%

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

      13. +-commutative 98.4%

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

      14. metadata-eval 98.4%

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

      15. sub-neg 98.4%

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

      16. div-sub 98.4%

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

      17. distribute-lft-neg-in 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in m around 0 98.4%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

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

   4. Taylor expanded in v around 0 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 0.1%

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

      4. mul-1-neg 0.1%

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

      5. unsub-neg 0.1%

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

   6. Simplified 0.1%

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

      1. div-inv 0.1%

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

      2. sub-neg 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 77.5%

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

      5. sqr-neg 77.5%

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

      6. sqrt-unprod 77.5%

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

      7. add-sqr-sqrt 77.5%

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

   8. Applied egg-rr 77.5%

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

   9. Taylor expanded in v around 0 77.5%

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

       1. associate-/l* 77.5%

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

       2. +-commutative 77.5%

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

   11. Simplified 77.5%

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

4. Final simplification 87.8%

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

Alternative 7: 87.4% 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 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 98.6%

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

      1. +-commutative 55.6%

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

      2. distribute-lft-in 55.6%

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

      3. div-inv 55.7%

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

      4. *-rgt-identity 55.7%

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

   7. Applied egg-rr 98.7%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

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

   4. Taylor expanded in v around 0 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 0.1%

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

      4. mul-1-neg 0.1%

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

      5. unsub-neg 0.1%

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

   6. Simplified 0.1%

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

      1. div-inv 0.1%

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

      2. sub-neg 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 77.5%

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

      5. sqr-neg 77.5%

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

      6. sqrt-unprod 77.5%

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

      7. add-sqr-sqrt 77.5%

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

   8. Applied egg-rr 77.5%

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

   9. Taylor expanded in v around 0 77.5%

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

       1. associate-/l* 77.5%

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

       2. +-commutative 77.5%

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

   11. Simplified 77.5%

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

4. Final simplification 87.9%

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

Alternative 8: 87.4% accurate, 1.1× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if m < 2.2999999999999998

   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 98.6%

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

      1. +-commutative 55.6%

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

      2. distribute-lft-in 55.6%

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

      3. div-inv 55.7%

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

      4. *-rgt-identity 55.7%

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

   7. Applied egg-rr 98.7%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

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

   4. Taylor expanded in v around 0 0.1%

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

      1. +-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 0.1%

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

      4. mul-1-neg 0.1%

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

      5. unsub-neg 0.1%

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

   6. Simplified 0.1%

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

      1. div-inv 0.1%

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

      2. sub-neg 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 77.5%

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

      5. sqr-neg 77.5%

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

      6. sqrt-unprod 77.5%

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

      7. add-sqr-sqrt 77.5%

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

   8. Applied egg-rr 77.5%

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

   9. Taylor expanded in v around 0 77.5%

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

4. Final simplification 87.9%

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

Alternative 9: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3.6e-184) -1.0 (+ m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.6e-184) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 3.6d-184) then
        tmp = -1.0d0
    else
        tmp = m + (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.6e-184) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.6e-184:
		tmp = -1.0
	else:
		tmp = m + (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.6e-184)
		tmp = -1.0;
	else
		tmp = Float64(m + Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.6e-184)
		tmp = -1.0;
	else
		tmp = m + (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.6e-184], -1.0, N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.6000000000000001e-184

   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 74.4%

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

   if 3.6000000000000001e-184 < 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 69.4%

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

   6. Taylor expanded in m around inf 61.3%

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

      1. +-commutative 61.3%

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

      2. distribute-lft-in 61.3%

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

      3. div-inv 61.4%

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

      4. *-rgt-identity 61.4%

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

   8. Applied egg-rr 61.4%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0 48.7%

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

4. Taylor expanded in v around 0 48.6%

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

   1. +-commutative 48.6%

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

   2. associate-*r* 48.6%

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

   3. distribute-rgt-out 48.6%

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

   4. mul-1-neg 48.6%

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

   5. unsub-neg 48.6%

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

6. Simplified 48.6%

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

   1. *-un-lft-identity 48.6%

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

   2. associate-/l* 48.6%

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

   3. sub-neg 48.6%

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

   4. add-sqr-sqrt 0.0%

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

   5. sqrt-unprod 87.9%

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

   6. sqr-neg 87.9%

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

   7. sqrt-unprod 87.9%

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

   8. add-sqr-sqrt 87.9%

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

8. Applied egg-rr 87.9%

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

   1. *-lft-identity 87.9%

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

10. Simplified 87.9%

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

11. Final simplification 87.9%

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

Alternative 11: 61.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.6e-184) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6e-184) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.6d-184) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.6e-184) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6e-184:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6e-184)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6e-184)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6e-184], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.59999999999999978e-184

   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 74.4%

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

   if 2.59999999999999978e-184 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 37.1%

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

   4. Taylor expanded in v around 0 37.1%

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

      1. +-commutative 37.1%

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

      2. associate-*r* 37.1%

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

      3. distribute-rgt-out 37.1%

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

      4. mul-1-neg 37.1%

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

      5. unsub-neg 37.1%

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

   6. Simplified 37.1%

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

   7. Taylor expanded in v around 0 29.0%

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

   8. Taylor expanded in m around 0 61.4%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{-32}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 5.2e-32) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.2e-32) {
		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 <= 5.2d-32) 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 <= 5.2e-32) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.2e-32:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.2e-32)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.2e-32)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.2e-32], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 5.1999999999999995e-32

   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 43.5%

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

   if 5.1999999999999995e-32 < 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 54.2%

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

   6. Taylor expanded in m around inf 54.2%

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

   7. Taylor expanded in v around inf 5.7%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{-32}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 22.8%

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

   1. neg-mul-1 22.8%

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

   2. sub-neg 22.8%

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

   3. +-commutative 22.8%

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

   4. distribute-neg-in 22.8%

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

   5. remove-double-neg 22.8%

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

   6. metadata-eval 22.8%

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

7. Simplified 22.8%

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

8. Final simplification 22.8%

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

Alternative 14: 24.8% 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 20.3%

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

6. Final simplification 20.3%

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

Reproduce

herbie shell --seed 2024071 
(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)))