b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 27.8s

Alternatives: 13

Speedup: 1.0×

• 42.1% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] * N[(m / 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. 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. *-commutative 99.9%

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

   3. associate-/l* 99.9%

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

   4. fma-neg 99.9%

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

   5. metadata-eval 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;-1 + \left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{\frac{v}{m}}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6e-8)
   (+ -1.0 (* (- 1.0 m) (/ m v)))
   (/ (- 1.0 m) (/ (/ v m) (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-8) {
		tmp = -1.0 + ((1.0 - m) * (m / v));
	} else {
		tmp = (1.0 - m) / ((v / m) / (1.0 - m));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-8) {
		tmp = -1.0 + ((1.0 - m) * (m / v));
	} else {
		tmp = (1.0 - m) / ((v / m) / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6e-8:
		tmp = -1.0 + ((1.0 - m) * (m / v))
	else:
		tmp = (1.0 - m) / ((v / m) / (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6e-8)
		tmp = Float64(-1.0 + Float64(Float64(1.0 - m) * Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) / Float64(Float64(v / m) / Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6e-8)
		tmp = -1.0 + ((1.0 - m) * (m / v));
	else
		tmp = (1.0 - m) / ((v / m) / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6e-8], N[(-1.0 + N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] / N[(N[(v / m), $MachinePrecision] / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001e-8

   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 99.5%

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

      1. *-rgt-identity 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. associate-*l/ 99.5%

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

      5. *-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 1.6000000000000001e-8 < m

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      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 v around 0 99.8%

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

   6. Taylor expanded in v around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. metadata-eval 99.4%

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

      3. distribute-neg-in 99.4%

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

      4. associate-/l* 99.4%

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

      5. sub0-neg 99.4%

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

      6. associate--r+ 99.4%

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

      7. metadata-eval 99.4%

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

   8. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-/r/ 99.4%

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

      3. clear-num 99.4%

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

      4. un-div-inv 99.4%

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

   10. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;-1 + \left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{\frac{v}{m}}{1 - m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;-1 + \left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6e-8)
   (+ -1.0 (* (- 1.0 m) (/ m v)))
   (* (- 1.0 m) (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-8) {
		tmp = -1.0 + ((1.0 - m) * (m / v));
	} else {
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-8) {
		tmp = -1.0 + ((1.0 - m) * (m / v));
	} else {
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6e-8:
		tmp = -1.0 + ((1.0 - m) * (m / v))
	else:
		tmp = (1.0 - m) * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6e-8)
		tmp = Float64(-1.0 + Float64(Float64(1.0 - m) * Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6e-8)
		tmp = -1.0 + ((1.0 - m) * (m / v));
	else
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6e-8], N[(-1.0 + N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001e-8

   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 99.5%

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

      1. *-rgt-identity 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. associate-*l/ 99.5%

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

      5. *-commutative 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 1.6000000000000001e-8 < m

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      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 v around 0 99.8%

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

   6. Taylor expanded in v around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. metadata-eval 99.4%

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

      3. distribute-neg-in 99.4%

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

      4. associate-/l* 99.4%

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

      5. sub0-neg 99.4%

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

      6. associate--r+ 99.4%

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

      7. metadata-eval 99.4%

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

   8. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;-1 + \left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(-1.0 + N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(N[(m + -1.0), $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 97.4%

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

      1. *-rgt-identity 97.4%

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

      2. sub-neg 97.4%

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

      3. metadata-eval 97.4%

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

      4. associate-*l/ 97.4%

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

      5. *-commutative 97.4%

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

   5. Applied egg-rr 97.4%

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

   if 1 < m

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      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 v around 0 99.8%

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

   6. Taylor expanded in v around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. associate-/l* 99.9%

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

      5. sub0-neg 99.9%

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

      6. associate--r+ 99.9%

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

      7. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in m around inf 97.6%

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

       1. neg-mul-1 97.6%

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

   11. Simplified 97.6%

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

4. Final simplification 97.5%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{-137}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3.2e-137) -1.0 (+ m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.2e-137) {
		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 <= 3.2d-137) 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 <= 3.2e-137) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.2e-137:
		tmp = -1.0
	else:
		tmp = m + (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.2e-137)
		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 <= 3.2e-137)
		tmp = -1.0;
	else
		tmp = m + (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.2e-137], -1.0, N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.20000000000000021e-137

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 77.2%

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

   if 3.20000000000000021e-137 < 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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 66.1%

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

   6. Taylor expanded in v around 0 66.3%

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

   7. Taylor expanded in m around inf 60.2%

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

      1. +-commutative 60.2%

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

   9. Simplified 60.2%

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

   10. Taylor expanded in v around inf 60.2%

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

Alternative 9: 61.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-137}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 5.5e-137) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.5e-137) {
		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 <= 5.5d-137) 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 <= 5.5e-137) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.5e-137:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.5e-137)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.5e-137)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.5e-137], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.5000000000000003e-137

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 77.2%

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

   if 5.5000000000000003e-137 < 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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 66.1%

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

   6. Taylor expanded in v around 0 66.3%

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

   7. Taylor expanded in m around 0 66.3%

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

      1. neg-mul-1 66.3%

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

   9. Simplified 66.3%

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

   10. Taylor expanded in m around inf 60.2%

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

Alternative 10: 26.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.1e-36) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-36) {
		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 <= 2.1d-36) 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 <= 2.1e-36) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.1e-36:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.1e-36)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.1e-36)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.1e-36], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 2.09999999999999991e-36

   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 51.2%

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

   if 2.09999999999999991e-36 < m

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      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 53.4%

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

   6. Taylor expanded in v around inf 5.0%

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

   7. Taylor expanded in m around inf 5.5%

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

Alternative 11: 75.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(m / v), $MachinePrecision] + -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 m around 0 75.0%

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

6. Taylor expanded in v around 0 75.1%

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

7. Final simplification 75.1%

   \[\leadsto \frac{m}{v} + -1 \]
8. Add Preprocessing

Alternative 12: 26.4% 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 26.5%

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

   1. neg-mul-1 26.5%

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

   2. sub-neg 26.5%

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

   3. +-commutative 26.5%

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

   4. distribute-neg-in 26.5%

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

   5. remove-double-neg 26.5%

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

   6. metadata-eval 26.5%

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

7. Simplified 26.5%

   \[\leadsto \color{blue}{m + -1} \]
8. Add Preprocessing

Alternative 13: 23.9% 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 24.2%

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

Reproduce

herbie shell --seed 2024116 
(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)))