b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.5s

Alternatives: 11

Speedup: 1.0×

• 40.8% 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, 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. fmm-def 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: 97.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0)
   (* (+ (/ m v) -1.0) (+ -1.0 (- 2.0 m)))
   (* (- 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) * (-1.0 + (2.0 - m));
	} 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)) * ((-1.0d0) + (2.0d0 - m))
    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) * (-1.0 + (2.0 - m));
	} 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) * (-1.0 + (2.0 - m))
	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(Float64(m / v) + -1.0) * Float64(-1.0 + Float64(2.0 - m)));
	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) * (-1.0 + (2.0 - m));
	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[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision] * N[(-1.0 + N[(2.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 - N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 98.2%

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

      1. expm1-log1p-u 98.2%

         \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]

   5. Applied egg-rr 98.2%

      \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-undefine 98.2%

         \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \]

      2. sub-neg 98.2%

         \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \]

      3. log1p-undefine 98.2%

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

      4. rem-exp-log 98.2%

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

      5. associate-+r- 98.2%

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

      6. metadata-eval 98.2%

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

      7. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around inf 97.3%

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

      1. neg-mul-1 97.3%

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

   7. Simplified 97.3%

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

4. Final simplification 97.8%

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

Alternative 3: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.42:\\ \;\;\;\;\left(\frac{m}{v} + -1\right) \cdot \left(-1 + \left(2 - m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.42)
   (* (+ (/ m v) -1.0) (+ -1.0 (- 2.0 m)))
   (* m (+ 1.0 (/ (* m m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.42) {
		tmp = ((m / v) + -1.0) * (-1.0 + (2.0 - m));
	} else {
		tmp = m * (1.0 + ((m * m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 0.42d0) then
        tmp = ((m / v) + (-1.0d0)) * ((-1.0d0) + (2.0d0 - m))
    else
        tmp = m * (1.0d0 + ((m * m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.42) {
		tmp = ((m / v) + -1.0) * (-1.0 + (2.0 - m));
	} else {
		tmp = m * (1.0 + ((m * m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.42:
		tmp = ((m / v) + -1.0) * (-1.0 + (2.0 - m))
	else:
		tmp = m * (1.0 + ((m * m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.42)
		tmp = Float64(Float64(Float64(m / v) + -1.0) * Float64(-1.0 + Float64(2.0 - m)));
	else
		tmp = Float64(m * Float64(1.0 + Float64(Float64(m * m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.42)
		tmp = ((m / v) + -1.0) * (-1.0 + (2.0 - m));
	else
		tmp = m * (1.0 + ((m * m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.42], N[(N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision] * N[(-1.0 + N[(2.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(1.0 + N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.419999999999999984

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 98.2%

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

      1. expm1-log1p-u 98.2%

         \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]

   5. Applied egg-rr 98.2%

      \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-undefine 98.2%

         \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \]

      2. sub-neg 98.2%

         \[\leadsto \left(\frac{m}{v} - 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \]

      3. log1p-undefine 98.2%

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

      4. rem-exp-log 98.2%

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

      5. associate-+r- 98.2%

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

      6. metadata-eval 98.2%

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

      7. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

   if 0.419999999999999984 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 97.3%

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

      1. neg-mul-1 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in m around inf 97.2%

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

      1. neg-mul-1 97.3%

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

   8. Simplified 97.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.42:\\ \;\;\;\;\left(\frac{m}{v} + -1\right) \cdot \left(-1 + \left(2 - m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / (1.0d0 - m))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / N[(v / N[(1.0 - m), $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. Taylor expanded in m around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. distribute-rgt-in 77.6%

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

   3. associate-/r/ 77.6%

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

   4. unpow-1 77.6%

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

   5. neg-mul-1 77.6%

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

   6. distribute-lft-neg-out 77.6%

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

   7. associate-/r/ 77.6%

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

   8. sub-neg 77.6%

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

   9. unpow-1 77.6%

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

   10. div-sub 99.8%

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

   11. associate-/r/ 99.8%

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

   12. associate-*l/ 99.9%

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

   13. associate-*r/ 99.9%

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

   14. *-commutative 99.9%

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

   15. associate-/r/ 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 5: 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 6: 87.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 98.2%

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

   4. Taylor expanded in m around 0 97.9%

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

   5. Taylor expanded in m around 0 97.9%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 0.1%

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

      1. div-inv 0.1%

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

      2. frac-2neg 0.1%

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

      3. neg-sub0 0.1%

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

      4. div-sub 0.1%

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

      5. add-sqr-sqrt 0.1%

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

      6. sqrt-unprod 0.1%

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

      7. sqr-neg 0.1%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 74.5%

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

      10. frac-2neg 74.5%

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

   7. Applied egg-rr 74.5%

      \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{\left(\frac{0}{-v} - \frac{m}{v}\right)} + -1\right) \]
   8. Step-by-step derivation

      1. div0 74.5%

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

      2. neg-sub0 74.5%

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

      3. distribute-frac-neg2 74.5%

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

   9. Simplified 74.5%

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

   10. Taylor expanded in v around 0 74.5%

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

       1. mul-1-neg 74.5%

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

       2. associate-/l* 74.5%

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

       3. distribute-rgt-neg-in 74.5%

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

   12. Simplified 74.5%

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

   13. Taylor expanded in m around 0 74.5%

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

       1. *-commutative 74.5%

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

       2. div-sub 74.5%

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

       3. sub-neg 74.5%

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

       4. metadata-eval 74.5%

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

       5. associate-*l/ 74.5%

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

       6. associate-*r/ 74.5%

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

       7. *-commutative 74.5%

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

   15. Simplified 74.5%

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

4. Final simplification 86.5%

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

Alternative 7: 87.2% 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. Add Preprocessing

3. Taylor expanded in m around 0 50.7%

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

   1. sub-neg 50.7%

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

   2. distribute-lft-in 50.7%

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

   3. *-commutative 50.7%

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

   4. *-un-lft-identity 50.7%

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

   5. sub-neg 50.7%

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

   6. metadata-eval 50.7%

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

   7. +-commutative 50.7%

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

   8. sub-neg 50.7%

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

   9. metadata-eval 50.7%

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

   10. +-commutative 50.7%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 86.5%

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

   13. sqr-neg 86.5%

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

   14. sqrt-unprod 86.5%

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

   15. add-sqr-sqrt 86.5%

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

5. Applied egg-rr 86.5%

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

   1. *-commutative 86.5%

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

   2. distribute-rgt1-in 86.5%

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

   3. +-commutative 86.5%

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

7. Simplified 86.5%

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

8. Final simplification 86.5%

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

Alternative 8: 26.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{-58}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3.5e-58) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.5e-58) {
		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 <= 3.5d-58) 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 <= 3.5e-58) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.5e-58:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.5e-58)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.5e-58)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.5e-58], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 3.4999999999999999e-58

   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 57.8%

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

   if 3.4999999999999999e-58 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 81.1%

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

      1. neg-mul-1 81.5%

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

   5. Simplified 81.1%

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

   6. Taylor expanded in m around 0 5.3%

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

Alternative 9: 76.1% 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. Add Preprocessing

3. Taylor expanded in m around 0 50.7%

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

4. Taylor expanded in m around 0 74.7%

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

5. Taylor expanded in m around 0 74.7%

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

6. Final simplification 74.7%

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

Alternative 10: 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 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 27.1%

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

   1. neg-mul-1 27.1%

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

   2. neg-sub0 27.1%

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

   3. associate--r- 27.1%

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

   4. metadata-eval 27.1%

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

7. Simplified 27.1%

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

8. Final simplification 27.1%

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

Alternative 11: 24.3% 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 24.8%

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

Reproduce

herbie shell --seed 2024170 
(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)))