b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.3%

Time: 7.9s

Alternatives: 14

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: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-41}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.5e-41)
   (+ -1.0 (+ m (/ m v)))
   (/ (* m (- 1.0 m)) (/ v (- 1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.5e-41) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (1.0 - 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 <= 6.5d-41) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m * (1.0d0 - m)) / (v / (1.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 6.5e-41) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (1.0 - m)) / (v / (1.0 - m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.5e-41:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m * (1.0 - m)) / (v / (1.0 - m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.5e-41)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(1.0 - m)) / Float64(v / Float64(1.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.5e-41)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m * (1.0 - m)) / (v / (1.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.5e-41], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.5000000000000004e-41

   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. associate-/l* 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. *-rgt-identity 99.9%

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

      6. associate-*r/ 100.0%

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

      7. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   if 6.5000000000000004e-41 < m

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. fma-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.8%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} \]
   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* 99.8%

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

      2. *-commutative 99.8%

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

      3. clear-num 99.8%

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

      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%

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

Alternative 2: 72.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{-200}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 9.2 \cdot 10^{-138}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + -1}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.8e-200)
   -1.0
   (if (<= m 1.2e-152)
     (/ m v)
     (if (<= m 9.2e-138)
       -1.0
       (if (<= m 1.0) (/ m v) (* m (/ (+ m -1.0) v)))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.8e-200) {
		tmp = -1.0;
	} else if (m <= 1.2e-152) {
		tmp = m / v;
	} else if (m <= 9.2e-138) {
		tmp = -1.0;
	} else if (m <= 1.0) {
		tmp = m / v;
	} else {
		tmp = m * ((m + -1.0) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 6.8d-200) then
        tmp = -1.0d0
    else if (m <= 1.2d-152) then
        tmp = m / v
    else if (m <= 9.2d-138) then
        tmp = -1.0d0
    else if (m <= 1.0d0) then
        tmp = m / v
    else
        tmp = m * ((m + (-1.0d0)) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 6.8e-200) {
		tmp = -1.0;
	} else if (m <= 1.2e-152) {
		tmp = m / v;
	} else if (m <= 9.2e-138) {
		tmp = -1.0;
	} else if (m <= 1.0) {
		tmp = m / v;
	} else {
		tmp = m * ((m + -1.0) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.8e-200:
		tmp = -1.0
	elif m <= 1.2e-152:
		tmp = m / v
	elif m <= 9.2e-138:
		tmp = -1.0
	elif m <= 1.0:
		tmp = m / v
	else:
		tmp = m * ((m + -1.0) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.8e-200)
		tmp = -1.0;
	elseif (m <= 1.2e-152)
		tmp = Float64(m / v);
	elseif (m <= 9.2e-138)
		tmp = -1.0;
	elseif (m <= 1.0)
		tmp = Float64(m / v);
	else
		tmp = Float64(m * Float64(Float64(m + -1.0) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.8e-200)
		tmp = -1.0;
	elseif (m <= 1.2e-152)
		tmp = m / v;
	elseif (m <= 9.2e-138)
		tmp = -1.0;
	elseif (m <= 1.0)
		tmp = m / v;
	else
		tmp = m * ((m + -1.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.8e-200], -1.0, If[LessEqual[m, 1.2e-152], N[(m / v), $MachinePrecision], If[LessEqual[m, 9.2e-138], -1.0, If[LessEqual[m, 1.0], N[(m / v), $MachinePrecision], N[(m * N[(N[(m + -1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < 6.8000000000000006e-200 or 1.2e-152 < m < 9.1999999999999996e-138

   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. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in m around 0 86.4%

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

   if 6.8000000000000006e-200 < m < 1.2e-152 or 9.1999999999999996e-138 < m < 1

   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* 99.7%

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

      3. fma-neg 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in v around 0 74.1%

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

      1. associate-*r/ 73.9%

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

   7. Simplified 73.9%

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

      1. associate-*r* 73.9%

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

      2. *-commutative 73.9%

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

      3. clear-num 73.9%

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

      4. un-div-inv 74.1%

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

   9. Applied egg-rr 74.1%

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

   10. Taylor expanded in m around 0 71.2%

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

   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. associate-/l* 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
   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 0 0.1%

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

      1. associate-*r* 0.1%

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

      2. *-commutative 0.1%

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

      3. div-inv 0.1%

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

      4. frac-2neg 0.1%

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

      5. distribute-rgt-neg-in 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 77.6%

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

      8. sqr-neg 77.6%

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

      9. sqrt-unprod 77.7%

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

      10. add-sqr-sqrt 77.7%

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

   10. Applied egg-rr 77.7%

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

       1. associate-/l* 77.7%

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

       2. neg-mul-1 77.7%

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

       3. *-commutative 77.7%

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

       4. *-commutative 77.7%

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

       5. neg-mul-1 77.7%

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

       6. neg-sub0 77.7%

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

       7. associate--r- 77.7%

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

       8. metadata-eval 77.7%

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

   12. Simplified 77.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{-200}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 9.2 \cdot 10^{-138}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + -1}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.3 \cdot 10^{-200}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{-152}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{-138}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 8.3e-200)
   -1.0
   (if (<= m 1.25e-152)
     (/ m v)
     (if (<= m 2.85e-138) -1.0 (* m (/ (+ m 1.0) v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.3e-200) {
		tmp = -1.0;
	} else if (m <= 1.25e-152) {
		tmp = m / v;
	} else if (m <= 2.85e-138) {
		tmp = -1.0;
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 8.3d-200) then
        tmp = -1.0d0
    else if (m <= 1.25d-152) then
        tmp = m / v
    else if (m <= 2.85d-138) then
        tmp = -1.0d0
    else
        tmp = m * ((m + 1.0d0) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 8.3e-200) {
		tmp = -1.0;
	} else if (m <= 1.25e-152) {
		tmp = m / v;
	} else if (m <= 2.85e-138) {
		tmp = -1.0;
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8.3e-200:
		tmp = -1.0
	elif m <= 1.25e-152:
		tmp = m / v
	elif m <= 2.85e-138:
		tmp = -1.0
	else:
		tmp = m * ((m + 1.0) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8.3e-200)
		tmp = -1.0;
	elseif (m <= 1.25e-152)
		tmp = Float64(m / v);
	elseif (m <= 2.85e-138)
		tmp = -1.0;
	else
		tmp = Float64(m * Float64(Float64(m + 1.0) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 8.3e-200)
		tmp = -1.0;
	elseif (m <= 1.25e-152)
		tmp = m / v;
	elseif (m <= 2.85e-138)
		tmp = -1.0;
	else
		tmp = m * ((m + 1.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 8.3e-200], -1.0, If[LessEqual[m, 1.25e-152], N[(m / v), $MachinePrecision], If[LessEqual[m, 2.85e-138], -1.0, N[(m * N[(N[(m + 1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < 8.3000000000000003e-200 or 1.2499999999999999e-152 < m < 2.8499999999999999e-138

   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. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in m around 0 86.4%

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

   if 8.3000000000000003e-200 < m < 1.2499999999999999e-152

   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. associate-/l* 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 73.2%

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

      1. associate-*r/ 73.1%

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

   7. Simplified 73.1%

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

      1. associate-*r* 73.1%

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

      2. *-commutative 73.1%

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

      3. clear-num 73.1%

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

      4. un-div-inv 73.2%

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

   9. Applied egg-rr 73.2%

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

   10. Taylor expanded in m around 0 73.2%

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

   if 2.8499999999999999e-138 < 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* 99.8%

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

      3. fma-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 92.3%

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

      1. associate-*r/ 92.2%

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

   7. Simplified 92.2%

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

   8. Taylor expanded in m around 0 21.1%

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

      1. un-div-inv 21.2%

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

      2. add-sqr-sqrt 21.0%

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

      3. sqrt-unprod 7.1%

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

      4. sqr-neg 7.1%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 55.2%

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

      7. clear-num 55.2%

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

      8. frac-2neg 55.2%

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

      9. metadata-eval 55.2%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 7.1%

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

      12. sqr-neg 7.1%

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

      13. sqrt-unprod 21.0%

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

      14. add-sqr-sqrt 21.1%

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

      15. distribute-frac-neg 21.1%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 55.1%

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

      18. sqr-neg 55.1%

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

      19. sqrt-unprod 55.2%

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

      20. add-sqr-sqrt 55.2%

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

   10. Applied egg-rr 55.2%

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

       1. associate-*r/ 55.2%

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

       2. frac-2neg 55.2%

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

       3. associate-/r/ 55.2%

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

       4. *-commutative 55.2%

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

       5. neg-mul-1 55.2%

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

       6. frac-2neg 55.2%

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

       7. sub-neg 55.2%

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

       8. add-sqr-sqrt 0.0%

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

       9. sqrt-unprod 0.8%

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

       10. sqr-neg 0.8%

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

       11. sqrt-unprod 0.8%

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

       12. add-sqr-sqrt 0.8%

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

       13. add-sqr-sqrt 0.0%

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

       14. sqrt-unprod 75.4%

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

       15. sqr-neg 75.4%

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

       16. sqrt-unprod 75.3%

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

       17. add-sqr-sqrt 75.4%

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

   12. Applied egg-rr 75.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.3 \cdot 10^{-200}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{-152}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{-138}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-14}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6e-14)
   (+ -1.0 (+ m (/ m v)))
   (* m (* (- 1.0 m) (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-14) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * ((1.0 - m) * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d-14) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = m * ((1.0d0 - m) * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-14) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * ((1.0 - m) * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6e-14:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = m * ((1.0 - m) * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6e-14)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(m * Float64(Float64(1.0 - m) * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6e-14)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = m * ((1.0 - m) * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6e-14], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(1.0 - m), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001e-14

   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. associate-/l* 99.8%

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

      3. fma-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. associate-*r/ 100.0%

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

      7. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   if 1.6000000000000001e-14 < m

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. fma-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.8%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} \]
   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. *-commutative 99.8%

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

      2. associate-*r* 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-14}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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. associate-/l* 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
   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 98.2%

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

       1. mul-1-neg 98.2%

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

       2. distribute-neg-frac2 98.2%

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

   12. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 6: 98.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} 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) * ((-1.0d0) + (m / v))
    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) * (-1.0 + (m / v));
	} 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) * (-1.0 + (m / v))
	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(-1.0 + Float64(m / v)));
	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) * (-1.0 + (m / v));
	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[(-1.0 + N[(m / v), $MachinePrecision]), $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 98.2%

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

   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. associate-/l* 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
   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 98.1%

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

      1. neg-mul-1 98.1%

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

      2. distribute-neg-frac2 98.1%

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

   10. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 7: 98.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} 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) * ((-1.0d0) + (m / v))
    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) * (-1.0 + (m / v));
	} 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) * (-1.0 + (m / v))
	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(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * 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 = (1.0 - m) * (-1.0 + (m / v));
	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[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m / v), $MachinePrecision] * N[(m + -1.0), $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%

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

   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. associate-/l* 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
   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. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in m around inf 98.1%

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

       1. neg-mul-1 98.1%

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

       2. distribute-neg-frac2 98.1%

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

   12. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 8: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 98.2%

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

   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. associate-/l* 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
   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 98.1%

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

      1. neg-mul-1 98.1%

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

      2. distribute-neg-frac2 98.1%

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

   10. Simplified 98.1%

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

       1. *-commutative 98.1%

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

       2. distribute-frac-neg2 98.1%

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

       3. distribute-frac-neg 98.1%

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

       4. associate-*l/ 98.1%

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

   12. Applied egg-rr 98.1%

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

       1. *-commutative 98.1%

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

       2. associate-/l* 98.1%

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

       3. associate-*l* 98.1%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 0.1%

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

       6. sqr-neg 0.1%

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

       7. sqrt-unprod 0.1%

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

       8. add-sqr-sqrt 0.1%

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

       9. clear-num 0.1%

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

       10. associate-/r/ 0.1%

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

       11. associate-/r* 0.1%

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

       12. clear-num 0.1%

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

       13. associate-/l* 0.1%

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

       14. sub-neg 0.1%

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

       15. add-sqr-sqrt 0.0%

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

       16. sqrt-unprod 98.0%

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

       17. sqr-neg 98.0%

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

       18. sqrt-unprod 98.0%

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

       19. add-sqr-sqrt 98.0%

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

   14. Applied egg-rr 98.0%

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

4. Final simplification 98.1%

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

Alternative 9: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.85:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m + 1}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.85) (+ -1.0 (+ m (/ m v))) (* m (* m (/ (+ m 1.0) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.85) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * (m * ((m + 1.0) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.85d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = m * (m * ((m + 1.0d0) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.85) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * (m * ((m + 1.0) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.85:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = m * (m * ((m + 1.0) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.85)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(m * Float64(m * Float64(Float64(m + 1.0) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.85)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = m * (m * ((m + 1.0) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.85], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(N[(m + 1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.85000000000000009

   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* 99.8%

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

      3. fma-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 97.3%

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

      1. sub-neg 97.3%

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

      2. metadata-eval 97.3%

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

      3. +-commutative 97.3%

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

      4. distribute-lft-in 97.3%

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

      5. *-rgt-identity 97.3%

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

      6. associate-*r/ 97.5%

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

      7. *-rgt-identity 97.5%

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

   7. Simplified 97.5%

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

   if 2.85000000000000009 < 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%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
   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 98.7%

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

      1. neg-mul-1 98.7%

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

      2. distribute-neg-frac2 98.7%

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

   10. Simplified 98.7%

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

       1. *-commutative 98.7%

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

       2. distribute-frac-neg2 98.7%

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

       3. distribute-frac-neg 98.7%

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

       4. associate-*l/ 98.7%

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

   12. Applied egg-rr 98.7%

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

       1. *-commutative 98.7%

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

       2. associate-/l* 98.7%

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

       3. associate-*l* 98.7%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 0.1%

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

       6. sqr-neg 0.1%

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

       7. sqrt-unprod 0.1%

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

       8. add-sqr-sqrt 0.1%

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

       9. clear-num 0.1%

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

       10. associate-/r/ 0.1%

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

       11. associate-/r* 0.1%

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

       12. clear-num 0.1%

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

       13. associate-/l* 0.1%

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

       14. sub-neg 0.1%

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

       15. add-sqr-sqrt 0.0%

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

       16. sqrt-unprod 98.7%

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

       17. sqr-neg 98.7%

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

       18. sqrt-unprod 98.6%

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

       19. add-sqr-sqrt 98.7%

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

   14. Applied egg-rr 98.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.85:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m + 1}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.9% accurate, 1.0× speedup

Math

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

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(m / Float64(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[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $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. associate-/l* 99.8%

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

   2. clear-num 99.8%

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

   3. un-div-inv 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 11: 87.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m + -1.0), $MachinePrecision] / v), $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. 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. associate-/l* 99.8%

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

      3. fma-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. metadata-eval 98.0%

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

      3. +-commutative 98.0%

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

      4. distribute-lft-in 98.0%

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

      5. *-rgt-identity 98.0%

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

      6. associate-*r/ 98.1%

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

      7. *-rgt-identity 98.1%

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

   7. Simplified 98.1%

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

   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. associate-/l* 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
   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 0 0.1%

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

      1. associate-*r* 0.1%

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

      2. *-commutative 0.1%

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

      3. div-inv 0.1%

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

      4. frac-2neg 0.1%

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

      5. distribute-rgt-neg-in 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 77.6%

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

      8. sqr-neg 77.6%

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

      9. sqrt-unprod 77.7%

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

      10. add-sqr-sqrt 77.7%

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

   10. Applied egg-rr 77.7%

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

       1. associate-/l* 77.7%

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

       2. neg-mul-1 77.7%

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

       3. *-commutative 77.7%

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

       4. *-commutative 77.7%

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

       5. neg-mul-1 77.7%

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

       6. neg-sub0 77.7%

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

       7. associate--r- 77.7%

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

       8. metadata-eval 77.7%

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

   12. Simplified 77.7%

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

4. Final simplification 87.7%

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

Alternative 12: 58.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 9 \cdot 10^{-106}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 9e-106) (/ m v) (+ m -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 9e-106) {
		tmp = m / v;
	} else {
		tmp = m + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 9d-106) then
        tmp = m / v
    else
        tmp = m + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 9e-106) {
		tmp = m / v;
	} else {
		tmp = m + -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 9e-106:
		tmp = m / v
	else:
		tmp = m + -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 9e-106)
		tmp = Float64(m / v);
	else
		tmp = Float64(m + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 9e-106)
		tmp = m / v;
	else
		tmp = m + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 9e-106], N[(m / v), $MachinePrecision], N[(m + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 8.99999999999999911e-106

   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* 99.8%

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

      3. fma-neg 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 84.5%

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

      1. associate-*r/ 84.4%

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

   7. Simplified 84.4%

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

      1. associate-*r* 84.4%

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

      2. *-commutative 84.4%

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

      3. clear-num 84.5%

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

      4. un-div-inv 84.5%

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

   9. Applied egg-rr 84.5%

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

   10. Taylor expanded in m around 0 67.4%

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

   if 8.99999999999999911e-106 < v

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around inf 45.5%

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

      1. neg-mul-1 45.5%

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

      2. neg-sub0 45.5%

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

      3. associate--r- 45.5%

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

      4. metadata-eval 45.5%

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

   7. Simplified 45.5%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 9 \cdot 10^{-106}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.6% 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. associate-/l* 99.8%

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

   3. fma-neg 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 26.1%

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

   1. neg-mul-1 26.1%

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

   2. neg-sub0 26.1%

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

   3. associate--r- 26.1%

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

   4. metadata-eval 26.1%

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

7. Simplified 26.1%

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

8. Final simplification 26.1%

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

Alternative 14: 24.2% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 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. associate-/l* 99.8%

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

   2. clear-num 99.8%

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

   3. un-div-inv 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in m around 0 23.7%

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

Reproduce

herbie shell --seed 2024096 
(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)))