b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Alternatives: 9

Speedup: N/A×

• 41.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. 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. *-lft-identity 99.9%

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

   3. *-lft-identity 99.9%

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

   4. sub-neg 99.9%

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

   5. *-commutative 99.9%

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

   6. associate-/l* 99.8%

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

   7. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. div-inv 99.8%

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

   2. clear-num 100.0%

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

   3. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(\left(1 - m\right) \cdot \left(1 - m\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.5e-20) (+ (/ m v) -1.0) (* (/ m v) (* (- 1.0 m) (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.5e-20) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m / v) * ((1.0 - 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 <= 6.5d-20) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = (m / v) * ((1.0d0 - m) * (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 6.5e-20) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m / v) * ((1.0 - m) * (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.5e-20:
		tmp = (m / v) + -1.0
	else:
		tmp = (m / v) * ((1.0 - m) * (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.5e-20)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(Float64(m / v) * Float64(Float64(1.0 - m) * Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.5e-20)
		tmp = (m / v) + -1.0;
	else
		tmp = (m / v) * ((1.0 - m) * (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.5e-20], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.50000000000000032e-20

   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. *-lft-identity 100.0%

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

      3. *-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 99.8%

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

   5. Taylor expanded in v around 0 99.8%

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

   6. Taylor expanded in m around 0 100.0%

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

   if 6.50000000000000032e-20 < 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. *-lft-identity 99.9%

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

      3. *-lft-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-/l* 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.6%

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

      1. associate-*l/ 99.6%

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

   6. Simplified 99.6%

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

      1. unpow2 99.6%

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

   8. Applied egg-rr 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(\left(1 - m\right) \cdot \left(1 - m\right)\right)\\ \end{array} \]

Alternative 3: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.45:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(m + -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.45) (+ (/ m v) -1.0) (* (/ m v) (* m (+ m -2.0)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.45) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m / v) * (m * (m + -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.45d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = (m / v) * (m * (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.45) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m / v) * (m * (m + -2.0));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.45:
		tmp = (m / v) + -1.0
	else:
		tmp = (m / v) * (m * (m + -2.0))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.45)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(Float64(m / v) * Float64(m * Float64(m + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.45)
		tmp = (m / v) + -1.0;
	else
		tmp = (m / v) * (m * (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.45], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.4500000000000002

   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. *-lft-identity 100.0%

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

      3. *-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 97.7%

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

   5. Taylor expanded in v around 0 97.7%

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

   6. Taylor expanded in m around 0 97.9%

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

   if 2.4500000000000002 < 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. *-lft-identity 99.9%

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

      3. *-lft-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-/l* 100.0%

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

      7. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-*l/ 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 99.3%

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

      1. +-commutative 99.3%

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

      2. unpow2 99.3%

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

      3. distribute-rgt-out 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.45:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(m + -2\right)\right)\\ \end{array} \]

Alternative 4: 87.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m / v) * (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 <= 2.4d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = (m / v) * (1.0d0 + m)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. *-lft-identity 100.0%

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

      3. *-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 97.7%

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

   5. Taylor expanded in v around 0 97.7%

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

   6. Taylor expanded in m around 0 97.9%

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

   if 2.39999999999999991 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l/ 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. distribute-lft-in 0.1%

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

      4. *-commutative 0.1%

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

      5. *-un-lft-identity 0.1%

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

      6. sub-neg 0.1%

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

      7. metadata-eval 0.1%

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

      8. sub-neg 0.1%

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

      9. metadata-eval 0.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 79.6%

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

      12. sqr-neg 79.6%

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

      13. sqrt-unprod 79.6%

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

      14. add-sqr-sqrt 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. distribute-rgt1-in 79.6%

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

   8. Simplified 79.6%

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

   9. Taylor expanded in v around 0 79.6%

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

       1. +-commutative 79.6%

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

       2. associate-*l/ 79.6%

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

   11. Simplified 79.6%

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

4. Final simplification 88.3%

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

Alternative 5: 87.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 + m), $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. associate-*l/ 100.0%

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

   3. fma-neg 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in m around 0 46.7%

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

   1. *-commutative 46.7%

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

   2. sub-neg 46.7%

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

   3. distribute-lft-in 46.7%

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

   4. *-commutative 46.7%

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

   5. *-un-lft-identity 46.7%

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

   6. sub-neg 46.7%

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

   7. metadata-eval 46.7%

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

   8. sub-neg 46.7%

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

   9. metadata-eval 46.7%

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

   10. add-sqr-sqrt 0.0%

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

   11. sqrt-unprod 88.3%

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

   12. sqr-neg 88.3%

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

   13. sqrt-unprod 88.3%

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

   14. add-sqr-sqrt 88.3%

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

6. Applied egg-rr 88.3%

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

   1. *-commutative 88.3%

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

   2. distribute-rgt1-in 88.3%

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

8. Simplified 88.3%

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

9. Final simplification 88.3%

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

Alternative 6: 62.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3 \cdot 10^{-166}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3e-166) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3e-166) {
		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 <= 3d-166) 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 <= 3e-166) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3e-166:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3e-166)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3e-166)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3e-166], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.0000000000000003e-166

   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. *-lft-identity 100.0%

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

      3. *-lft-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 79.2%

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

   if 3.0000000000000003e-166 < 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. *-lft-identity 99.9%

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

      3. *-lft-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-/l* 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in v around 0 88.5%

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

      1. associate-*l/ 88.5%

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

   6. Simplified 88.5%

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

   7. Taylor expanded in m around 0 54.0%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3 \cdot 10^{-166}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 7: 76.2% 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. *-lft-identity 99.9%

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

   3. *-lft-identity 99.9%

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

   4. sub-neg 99.9%

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

   5. *-commutative 99.9%

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

   6. associate-/l* 99.8%

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

   7. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in m around 0 72.9%

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

5. Taylor expanded in v around 0 72.9%

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

6. Taylor expanded in m around 0 72.9%

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

7. Final simplification 72.9%

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

Alternative 8: 28.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. *-lft-identity 99.9%

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

   3. *-lft-identity 99.9%

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

   4. sub-neg 99.9%

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

   5. *-commutative 99.9%

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

   6. associate-/l* 99.8%

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

   7. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around inf 28.3%

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

   1. mul-1-neg 28.3%

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

   2. sub0-neg 28.3%

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

   3. associate--r- 28.3%

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

   4. metadata-eval 28.3%

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

   5. +-commutative 28.3%

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

6. Simplified 28.3%

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

7. Final simplification 28.3%

   \[\leadsto m + -1 \]

Alternative 9: 25.6% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 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. *-lft-identity 99.9%

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

   3. *-lft-identity 99.9%

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

   4. sub-neg 99.9%

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

   5. *-commutative 99.9%

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

   6. associate-/l* 99.8%

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

   7. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in m around 0 25.7%

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

5. Final simplification 25.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023336 
(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)))