b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 10

Speedup: 1.0×

• 41.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 96.3%

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

      1. *-commutative 96.3%

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

      2. sub-neg 96.3%

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

      3. metadata-eval 96.3%

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

      4. distribute-lft-in 96.3%

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

      5. *-commutative 96.3%

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

      6. neg-mul-1 96.3%

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

   4. Applied egg-rr 96.3%

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

   5. Taylor expanded in v around 0 96.3%

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

      1. sub-neg 96.3%

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

      2. +-commutative 96.3%

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

      3. metadata-eval 96.3%

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

      4. associate-+l+ 96.3%

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

      5. associate-/l* 96.3%

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

      6. +-commutative 96.3%

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

   7. Simplified 96.3%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around inf 98.5%

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

      1. associate-*r/ 98.5%

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

      2. neg-mul-1 98.5%

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

   6. Simplified 98.5%

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

      1. frac-2neg 98.5%

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

      2. remove-double-neg 98.5%

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

      3. associate-/r/ 98.5%

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

   8. Applied egg-rr 98.5%

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

4. Final simplification 97.3%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. sub-neg 100.0%

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

   3. associate-*l/ 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 37.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 4.6 \cdot 10^{-152}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= v 4.6e-152) (* (- 1.0 m) (/ m v)) (+ m -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 4.6e-152) {
		tmp = (1.0 - m) * (m / v);
	} else {
		tmp = m + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 4.6d-152) then
        tmp = (1.0d0 - m) * (m / v)
    else
        tmp = m + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 4.6e-152) {
		tmp = (1.0 - m) * (m / v);
	} else {
		tmp = m + -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 4.6e-152:
		tmp = (1.0 - m) * (m / v)
	else:
		tmp = m + -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 4.6e-152)
		tmp = Float64(Float64(1.0 - m) * Float64(m / v));
	else
		tmp = Float64(m + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 4.6e-152)
		tmp = (1.0 - m) * (m / v);
	else
		tmp = m + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 4.6e-152], N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 4.6000000000000003e-152

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 55.2%

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

      1. *-commutative 55.2%

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

      2. sub-neg 55.2%

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

      3. metadata-eval 55.2%

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

      4. distribute-lft-in 55.2%

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

      5. *-commutative 55.2%

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

      6. neg-mul-1 55.2%

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

   4. Applied egg-rr 55.2%

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

   5. Taylor expanded in v around 0 55.2%

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

      1. sub-neg 55.2%

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

      2. +-commutative 55.2%

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

      3. metadata-eval 55.2%

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

      4. associate-+l+ 55.2%

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

      5. associate-/l* 55.2%

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

      6. +-commutative 55.2%

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

   7. Simplified 55.2%

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

   8. Taylor expanded in v around 0 45.0%

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

      1. associate-*l/ 45.0%

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

      2. *-commutative 45.0%

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

   10. Simplified 45.0%

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

   if 4.6000000000000003e-152 < v

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around inf 43.0%

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

      1. neg-mul-1 43.0%

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

      2. neg-sub0 43.0%

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

      3. associate--r- 43.0%

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

      4. metadata-eval 43.0%

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

   6. Simplified 43.0%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 4.6 \cdot 10^{-152}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 5: 71.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-223}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.9e-223) -1.0 (* (/ m v) (+ m 1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-223) {
		tmp = -1.0;
	} else {
		tmp = (m / v) * (m + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.9d-223) then
        tmp = -1.0d0
    else
        tmp = (m / v) * (m + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.9e-223) {
		tmp = -1.0;
	} else {
		tmp = (m / v) * (m + 1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.9e-223:
		tmp = -1.0
	else:
		tmp = (m / v) * (m + 1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.9e-223)
		tmp = -1.0;
	else
		tmp = Float64(Float64(m / v) * Float64(m + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.9e-223)
		tmp = -1.0;
	else
		tmp = (m / v) * (m + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.9e-223], -1.0, N[(N[(m / v), $MachinePrecision] * N[(m + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.9e-223

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 83.0%

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

   if 2.9e-223 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 44.4%

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

      1. *-commutative 44.4%

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

      2. sub-neg 44.4%

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

      3. metadata-eval 44.4%

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

      4. distribute-lft-in 44.4%

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

      5. *-commutative 44.4%

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

      6. neg-mul-1 44.4%

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

   4. Applied egg-rr 44.4%

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

      1. unsub-neg 44.4%

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

      2. *-commutative 44.4%

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

      3. *-un-lft-identity 44.4%

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

      4. distribute-rgt-out-- 44.4%

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

      5. sub-neg 44.4%

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

      6. +-commutative 44.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 87.0%

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

      9. sqr-neg 87.0%

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

      10. sqrt-unprod 87.0%

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

      11. add-sqr-sqrt 87.0%

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

   6. Applied egg-rr 87.0%

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

   7. Taylor expanded in v around 0 72.1%

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

      1. +-commutative 72.1%

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

      2. associate-*l/ 72.1%

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

   9. Simplified 72.1%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-223}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + 1\right)\\ \end{array} \]

Alternative 6: 87.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 95.9%

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

   5. Taylor expanded in v around 0 95.9%

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

   if 2.39999999999999991 < m

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

      4. distribute-lft-in 0.0%

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

      5. *-commutative 0.0%

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

      6. neg-mul-1 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. unsub-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. *-un-lft-identity 0.0%

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

      4. distribute-rgt-out-- 0.0%

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

      5. sub-neg 0.0%

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

      6. +-commutative 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 80.3%

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

      9. sqr-neg 80.3%

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

      10. sqrt-unprod 80.3%

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

      11. add-sqr-sqrt 80.3%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in v around 0 80.3%

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

      1. +-commutative 80.3%

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

      2. associate-*l/ 80.3%

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

   9. Simplified 80.3%

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

4. Final simplification 88.9%

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

Alternative 7: 87.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 95.9%

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

      1. +-commutative 95.9%

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

      2. distribute-lft-in 95.9%

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

      3. div-inv 96.2%

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

      4. *-rgt-identity 96.2%

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

   6. Applied egg-rr 96.2%

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

   if 2.39999999999999991 < m

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

      4. distribute-lft-in 0.0%

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

      5. *-commutative 0.0%

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

      6. neg-mul-1 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. unsub-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. *-un-lft-identity 0.0%

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

      4. distribute-rgt-out-- 0.0%

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

      5. sub-neg 0.0%

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

      6. +-commutative 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 80.3%

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

      9. sqr-neg 80.3%

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

      10. sqrt-unprod 80.3%

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

      11. add-sqr-sqrt 80.3%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in v around 0 80.3%

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

      1. +-commutative 80.3%

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

      2. associate-*l/ 80.3%

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

   9. Simplified 80.3%

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

4. Final simplification 89.0%

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

Alternative 8: 87.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in m around 0 53.0%

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

   1. *-commutative 53.0%

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

   2. sub-neg 53.0%

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

   3. metadata-eval 53.0%

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

   4. distribute-lft-in 53.0%

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

   5. *-commutative 53.0%

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

   6. neg-mul-1 53.0%

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

4. Applied egg-rr 53.0%

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

   1. unsub-neg 53.0%

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

   2. *-commutative 53.0%

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

   3. *-un-lft-identity 53.0%

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

   4. distribute-rgt-out-- 53.0%

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

   5. sub-neg 53.0%

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

   6. +-commutative 53.0%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 89.0%

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

   9. sqr-neg 89.0%

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

   10. sqrt-unprod 89.0%

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

   11. add-sqr-sqrt 89.0%

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

6. Applied egg-rr 89.0%

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

7. Final simplification 89.0%

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

Alternative 9: 26.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. sub-neg 100.0%

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

   3. associate-*l/ 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around inf 26.1%

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

   1. neg-mul-1 26.1%

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

   2. neg-sub0 26.1%

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

   3. associate--r- 26.1%

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

   4. metadata-eval 26.1%

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

6. Simplified 26.1%

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

7. Final simplification 26.1%

   \[\leadsto m + -1 \]

Alternative 10: 24.2% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. sub-neg 100.0%

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

   3. associate-*l/ 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in m around 0 23.9%

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

5. Final simplification 23.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023320 
(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)))