b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.5%

Time: 6.1s

Alternatives: 11

Speedup: N/A×

• 41.4% 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: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m \cdot \left(2 - m\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6e-30) (+ (/ m v) -1.0) (/ (* m (- 1.0 (* m (- 2.0 m)))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-30) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m * (1.0 - (m * (2.0 - m)))) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d-30) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = (m * (1.0d0 - (m * (2.0d0 - m)))) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-30) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (m * (1.0 - (m * (2.0 - m)))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6e-30:
		tmp = (m / v) + -1.0
	else:
		tmp = (m * (1.0 - (m * (2.0 - m)))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6e-30)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(Float64(m * Float64(1.0 - Float64(m * Float64(2.0 - m)))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6e-30)
		tmp = (m / v) + -1.0;
	else
		tmp = (m * (1.0 - (m * (2.0 - m)))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6e-30], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(m * N[(1.0 - N[(m * N[(2.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6e-30

   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 0 100.0%

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

   4. Taylor expanded in m around 0 100.0%

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

   if 1.6e-30 < 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. *-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. Taylor expanded in v around 0 99.9%

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

      1. associate-*r/ 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 99.9%

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

      2. associate-*r/ 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in m around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m \cdot \left(2 - m\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7.8e-17) (+ (/ m v) -1.0) (* (- 1.0 m) (/ (- 1.0 m) (/ v m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.8e-17) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (1.0 - m) * ((1.0 - m) / (v / 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 <= 7.8d-17) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = (1.0d0 - m) * ((1.0d0 - m) / (v / m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7.8e-17) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (1.0 - m) * ((1.0 - m) / (v / m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.8e-17:
		tmp = (m / v) + -1.0
	else:
		tmp = (1.0 - m) * ((1.0 - m) / (v / m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.8e-17)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(1.0 - m) / Float64(v / m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7.8e-17)
		tmp = (m / v) + -1.0;
	else
		tmp = (1.0 - m) * ((1.0 - m) / (v / m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.8e-17], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.79999999999999979e-17

   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 0 100.0%

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

   4. Taylor expanded in m around 0 100.0%

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

   if 7.79999999999999979e-17 < 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. *-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. Taylor expanded in v around 0 99.9%

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

      1. associate-*r/ 99.9%

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

   7. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. add-sqr-sqrt 99.8%

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

      3. times-frac 99.8%

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

      4. metadata-eval 99.8%

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

      5. sqrt-div 99.9%

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

      6. inv-pow 99.9%

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

      7. metadata-eval 99.9%

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

      8. sqrt-pow1 99.9%

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

      9. metadata-eval 99.9%

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

      10. metadata-eval 99.9%

          \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \left({v}^{\color{blue}{-0.5}} \cdot \frac{1 - m}{\sqrt{v}}\right)\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \color{blue}{\left({v}^{-0.5} \cdot \frac{1 - m}{\sqrt{v}}\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r/ 99.9%

          \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \color{blue}{\frac{{v}^{-0.5} \cdot \left(1 - m\right)}{\sqrt{v}}}\right) \]

       2. *-commutative 99.9%

          \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{\color{blue}{\left(1 - m\right) \cdot {v}^{-0.5}}}{\sqrt{v}}\right) \]

       3. associate-/l* 99.9%

          \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \color{blue}{\left(\left(1 - m\right) \cdot \frac{{v}^{-0.5}}{\sqrt{v}}\right)}\right) \]

   11. Simplified 99.9%

       \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \color{blue}{\left(\left(1 - m\right) \cdot \frac{{v}^{-0.5}}{\sqrt{v}}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-*r* 99.9%

          \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{{v}^{-0.5}}{\sqrt{v}}\right)} \]

       2. pow1/2 99.9%

          \[\leadsto \left(1 - m\right) \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{{v}^{-0.5}}{\color{blue}{{v}^{0.5}}}\right) \]

       3. pow-div 99.9%

          \[\leadsto \left(1 - m\right) \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot \color{blue}{{v}^{\left(-0.5 - 0.5\right)}}\right) \]

       4. metadata-eval 99.9%

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

       5. inv-pow 99.9%

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

       6. div-inv 99.9%

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

       7. *-commutative 99.9%

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

       8. associate-*l/ 99.9%

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

       9. associate-/r/ 99.9%

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

   13. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 9.2e-19) (+ (/ m v) -1.0) (* (- 1.0 m) (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 9.2e-19) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 9.2d-19) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = (1.0d0 - m) * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 9.2e-19) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 9.2e-19:
		tmp = (m / v) + -1.0
	else:
		tmp = (1.0 - m) * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 9.2e-19)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 9.2e-19)
		tmp = (m / v) + -1.0;
	else
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 9.2e-19], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.19999999999999919e-19

   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 0 100.0%

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

   4. Taylor expanded in m around 0 100.0%

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

   if 9.19999999999999919e-19 < 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. *-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. Taylor expanded in v around 0 99.9%

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

      1. associate-*r/ 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m / v), $MachinePrecision]), $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 99.3%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. *-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. Taylor expanded in v around 0 99.9%

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

      1. associate-*r/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf 96.8%

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

      1. neg-mul-1 96.8%

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

   10. Simplified 96.8%

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

4. Final simplification 98.1%

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

Alternative 5: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.43) (* (- 1.0 m) (+ (/ m v) -1.0)) (/ (* m m) (/ v m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m * m) / (v / 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 <= 0.43d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = (m * m) / (v / m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m * m) / (v / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.43:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = (m * m) / (v / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.43)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m * m) / Float64(v / m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.43)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = (m * m) / (v / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.43], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m * m), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.429999999999999993

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 99.3%

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

   if 0.429999999999999993 < 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. *-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. Taylor expanded in v around 0 99.9%

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

      1. associate-*r/ 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in m around inf 96.8%

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

       1. neg-mul-1 96.8%

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

   12. Simplified 96.8%

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

   13. Taylor expanded in m around inf 96.7%

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

       1. associate-*r/ 96.7%

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

       2. neg-mul-1 96.7%

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

   15. Simplified 96.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. associate-*r/ 99.9%

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

   3. *-commutative 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 7: 62.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.5 \cdot 10^{-169}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 9.5e-169) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 9.5e-169) {
		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 <= 9.5d-169) 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 <= 9.5e-169) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 9.5e-169:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 9.5e-169)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 9.5e-169)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 9.5e-169], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.5000000000000001e-169

   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. *-commutative 100.0%

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

      3. associate-/l* 100.0%

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

      4. fmm-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 78.8%

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

   if 9.5000000000000001e-169 < 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. *-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. Taylor expanded in m around 0 67.7%

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

      1. +-commutative 67.7%

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

      2. distribute-lft-in 67.7%

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

      3. div-inv 67.8%

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

      4. *-rgt-identity 67.8%

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

   7. Applied egg-rr 67.8%

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

   8. Taylor expanded in v around 0 59.9%

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

Alternative 8: 26.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-31}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 9.6e-31) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 9.6e-31) {
		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 <= 9.6d-31) 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 <= 9.6e-31) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 9.6e-31:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 9.6e-31)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 9.6e-31)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 9.6e-31], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 9.6000000000000001e-31

   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. *-commutative 100.0%

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

      3. associate-/l* 100.0%

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

      4. fmm-def 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 55.2%

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

   if 9.6000000000000001e-31 < 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. *-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. Taylor expanded in v around inf 5.4%

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

      1. mul-1-neg 5.4%

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

      2. sub-neg 5.4%

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

      3. distribute-neg-in 5.4%

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

      4. metadata-eval 5.4%

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

      5. remove-double-neg 5.4%

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

   7. Simplified 5.4%

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

   8. Taylor expanded in m around inf 5.7%

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

Alternative 9: 75.2% 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. Add Preprocessing

3. Taylor expanded in m around 0 52.0%

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

4. Taylor expanded in m around 0 76.6%

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

5. Final simplification 76.6%

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

Alternative 10: 26.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. *-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. Taylor expanded in v around inf 29.5%

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

   1. mul-1-neg 29.5%

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

   2. sub-neg 29.5%

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

   3. distribute-neg-in 29.5%

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

   4. metadata-eval 29.5%

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

   5. remove-double-neg 29.5%

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

7. Simplified 29.5%

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

8. Final simplification 29.5%

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

Alternative 11: 23.9% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. *-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. Taylor expanded in m around 0 27.1%

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