b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.4%

Time: 8.8s

Alternatives: 13

Speedup: N/A×

• 41.1% 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-33}:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + m \cdot \left(m + -2\right)}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.25e-33)
   (+ m (+ -1.0 (/ m v)))
   (/ (+ 1.0 (* m (+ m -2.0))) (/ v m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-33) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (1.0 + (m * (m + -2.0))) / (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 <= 1.25d-33) then
        tmp = m + ((-1.0d0) + (m / v))
    else
        tmp = (1.0d0 + (m * (m + (-2.0d0)))) / (v / m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-33) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (1.0 + (m * (m + -2.0))) / (v / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.25e-33:
		tmp = m + (-1.0 + (m / v))
	else:
		tmp = (1.0 + (m * (m + -2.0))) / (v / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.25e-33)
		tmp = Float64(m + Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 + Float64(m * Float64(m + -2.0))) / Float64(v / m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.25e-33)
		tmp = m + (-1.0 + (m / v));
	else
		tmp = (1.0 + (m * (m + -2.0))) / (v / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.25e-33], N[(m + N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.25000000000000007e-33

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. *-lft-identity 99.8%

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

      3. associate--l+ 99.8%

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

      4. associate-*l/ 100.0%

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

      5. *-lft-identity 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if 1.25000000000000007e-33 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. fma-def 99.9%

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

      3. clear-num 99.9%

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

      4. metadata-eval 99.9%

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

      5. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in v around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around 0 99.9%

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

       1. unpow2 99.9%

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

       2. distribute-rgt-in 99.9%

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

       3. +-commutative 99.9%

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

   13. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-33}:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + m \cdot \left(m + -2\right)}{\frac{v}{m}}\\ \end{array} \]

Alternative 2: 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. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 97.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.65:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1}{\frac{v}{m \cdot m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.65) (+ m (+ -1.0 (/ m v))) (* m (/ 1.0 (/ v (* m m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.65) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = m * (1.0 / (v / (m * m)));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.65d0) then
        tmp = m + ((-1.0d0) + (m / v))
    else
        tmp = m * (1.0d0 / (v / (m * m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.65) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = m * (1.0 / (v / (m * m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.65:
		tmp = m + (-1.0 + (m / v))
	else:
		tmp = m * (1.0 / (v / (m * m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.65)
		tmp = Float64(m + Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(1.0 / Float64(v / Float64(m * m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.65)
		tmp = m + (-1.0 + (m / v));
	else
		tmp = m * (1.0 / (v / (m * m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.65], N[(m + N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(1.0 / N[(v / N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.64999999999999991

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 97.7%

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

      1. distribute-rgt-in 97.7%

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

      2. *-lft-identity 97.7%

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

      3. associate--l+ 97.7%

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

      4. associate-*l/ 97.8%

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

      5. *-lft-identity 97.8%

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

      6. sub-neg 97.8%

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

      7. metadata-eval 97.8%

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

   6. Simplified 97.8%

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

   if 2.64999999999999991 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in v around 0 98.6%

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

      1. mul-1-neg 98.6%

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

      2. associate-/l* 98.6%

         \[\leadsto -\color{blue}{\frac{{m}^{2}}{\frac{v}{1 - m}}} \]

      3. distribute-neg-frac 98.6%

         \[\leadsto \color{blue}{\frac{-{m}^{2}}{\frac{v}{1 - m}}} \]

      4. unpow2 98.6%

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

      5. distribute-rgt-neg-in 98.6%

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

   9. Simplified 98.6%

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

       1. associate-/l* 98.6%

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

       2. div-inv 98.6%

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

       3. sub-neg 98.6%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 0.1%

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

       6. sqr-neg 0.1%

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

       7. sqrt-unprod 0.1%

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

       8. add-sqr-sqrt 0.1%

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

       9. add-sqr-sqrt 0.0%

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

       10. sqrt-unprod 98.5%

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

       11. sqr-neg 98.5%

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

       12. sqrt-unprod 98.5%

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

       13. add-sqr-sqrt 98.5%

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

   11. Applied egg-rr 98.5%

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

   12. Taylor expanded in m around inf 98.6%

       \[\leadsto m \cdot \frac{1}{\color{blue}{\frac{v}{{m}^{2}}}} \]
   13. Step-by-step derivation

       1. unpow2 98.6%

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

   14. Simplified 98.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.65:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1}{\frac{v}{m \cdot m}}\\ \end{array} \]

Alternative 4: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (+ m (+ -1.0 (/ m v))) (* (+ m -2.0) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.4d0) then
        tmp = m + ((-1.0d0) + (m / v))
    else
        tmp = (m + (-2.0d0)) * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = m + (-1.0 + (m / v))
	else:
		tmp = (m + -2.0) * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(m + Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(m + -2.0) * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = m + (-1.0 + (m / v));
	else
		tmp = (m + -2.0) * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(m + N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m + -2.0), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 97.7%

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

      1. distribute-rgt-in 97.7%

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

      2. *-lft-identity 97.7%

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

      3. associate--l+ 97.7%

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

      4. associate-*l/ 97.8%

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

      5. *-lft-identity 97.8%

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

      6. sub-neg 97.8%

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

      7. metadata-eval 97.8%

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

   6. Simplified 97.8%

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

   if 2.39999999999999991 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. fma-def 99.9%

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

      3. clear-num 99.9%

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

      4. metadata-eval 99.9%

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

      5. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in m around inf 21.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   9. Step-by-step derivation

      1. unpow2 21.4%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. cube-mult 21.3%

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

      3. associate-*r/ 21.3%

         \[\leadsto -2 \cdot \frac{m \cdot m}{v} + \color{blue}{m \cdot \frac{m \cdot m}{v}} \]

      4. distribute-rgt-out 99.5%

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

      5. associate-*r/ 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 5: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.62:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.62) (* (- 1.0 m) (+ -1.0 (/ m v))) (* (+ m -2.0) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.62) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.62d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (m / v))
    else
        tmp = (m + (-2.0d0)) * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.62) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.62:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = (m + -2.0) * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.62)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(m + -2.0) * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.62)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	else
		tmp = (m + -2.0) * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.62], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m + -2.0), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6200000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 97.9%

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

   if 1.6200000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. fma-def 99.9%

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

      3. clear-num 99.9%

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

      4. metadata-eval 99.9%

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

      5. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in m around inf 21.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   9. Step-by-step derivation

      1. unpow2 21.4%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. cube-mult 21.3%

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

      3. associate-*r/ 21.3%

         \[\leadsto -2 \cdot \frac{m \cdot m}{v} + \color{blue}{m \cdot \frac{m \cdot m}{v}} \]

      4. distribute-rgt-out 99.5%

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

      5. associate-*r/ 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.62:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 6: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.62:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m + -2\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.62) (* (- 1.0 m) (+ -1.0 (/ m v))) (/ (* (+ m -2.0) (* m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.62) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = ((m + -2.0) * (m * m)) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.62d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (m / v))
    else
        tmp = ((m + (-2.0d0)) * (m * m)) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.62) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = ((m + -2.0) * (m * m)) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.62:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = ((m + -2.0) * (m * m)) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.62)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(Float64(m + -2.0) * Float64(m * m)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.62)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	else
		tmp = ((m + -2.0) * (m * m)) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.62], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m + -2.0), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6200000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 97.9%

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

   if 1.6200000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. fma-def 99.9%

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

      3. clear-num 99.9%

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

      4. metadata-eval 99.9%

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

      5. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in m around inf 21.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   9. Step-by-step derivation

      1. unpow2 21.4%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. cube-mult 21.3%

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

      3. associate-*r/ 21.3%

         \[\leadsto -2 \cdot \frac{m \cdot m}{v} + \color{blue}{m \cdot \frac{m \cdot m}{v}} \]

      4. distribute-rgt-out 99.5%

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

      5. associate-*r/ 99.5%

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

   10. Simplified 99.5%

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

       1. *-commutative 99.5%

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

       2. associate-*r/ 99.5%

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

       3. associate-*r/ 99.5%

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

       4. +-commutative 99.5%

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

   12. Applied egg-rr 99.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.62:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m + -2\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

Alternative 7: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6) (+ m (+ -1.0 (/ m v))) (/ (* m m) (/ v m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = m + (-1.0 + (m / v));
	} 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 <= 2.6d0) then
        tmp = m + ((-1.0d0) + (m / v))
    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 <= 2.6) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (m * m) / (v / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6:
		tmp = m + (-1.0 + (m / v))
	else:
		tmp = (m * m) / (v / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6)
		tmp = Float64(m + Float64(-1.0 + Float64(m / v)));
	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 <= 2.6)
		tmp = m + (-1.0 + (m / v));
	else
		tmp = (m * m) / (v / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6], N[(m + N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * m), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000009

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 97.7%

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

      1. distribute-rgt-in 97.7%

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

      2. *-lft-identity 97.7%

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

      3. associate--l+ 97.7%

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

      4. associate-*l/ 97.8%

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

      5. *-lft-identity 97.8%

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

      6. sub-neg 97.8%

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

      7. metadata-eval 97.8%

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

   6. Simplified 97.8%

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

   if 2.60000000000000009 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. fma-def 99.9%

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

      3. clear-num 99.9%

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

      4. metadata-eval 99.9%

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

      5. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in v around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf 98.6%

       \[\leadsto \frac{\color{blue}{{m}^{2}}}{\frac{v}{m}} \]
   12. Step-by-step derivation

       1. unpow2 98.6%

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

   13. Simplified 98.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]

Alternative 8: 62.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.75 \cdot 10^{-172}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.75e-172) -1.0 (+ m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.75e-172) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.75d-172) then
        tmp = -1.0d0
    else
        tmp = m + (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.75e-172) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.75e-172:
		tmp = -1.0
	else:
		tmp = m + (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.75e-172)
		tmp = -1.0;
	else
		tmp = Float64(m + Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.75e-172)
		tmp = -1.0;
	else
		tmp = m + (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.75e-172], -1.0, N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.75000000000000014e-172

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 75.3%

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

   if 1.75000000000000014e-172 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 68.5%

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

      1. distribute-rgt-in 68.5%

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

      2. *-lft-identity 68.5%

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

      3. associate--l+ 68.5%

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

      4. associate-*l/ 68.5%

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

      5. *-lft-identity 68.5%

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

      6. sub-neg 68.5%

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

      7. metadata-eval 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in m around inf 60.5%

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

      1. distribute-rgt-in 60.5%

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

      2. associate-*l/ 60.5%

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

      3. associate-*r/ 60.5%

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

      4. *-lft-identity 60.5%

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

      5. *-lft-identity 60.5%

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

   9. Simplified 60.5%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.75 \cdot 10^{-172}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \]

Alternative 9: 76.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 76.1%

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

   1. distribute-rgt-in 76.1%

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

   2. *-lft-identity 76.1%

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

   3. associate--l+ 76.1%

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

   4. associate-*l/ 76.2%

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

   5. *-lft-identity 76.2%

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

   6. sub-neg 76.2%

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

   7. metadata-eval 76.2%

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

6. Simplified 76.2%

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

7. Final simplification 76.2%

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

Alternative 10: 62.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-170}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.6e-170) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6e-170) {
		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 <= 2.6d-170) 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 <= 2.6e-170) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6e-170:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6e-170)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6e-170)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6e-170], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.6000000000000001e-170

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 75.3%

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

   if 2.6000000000000001e-170 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.8%

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

      2. fma-def 99.8%

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

      3. clear-num 99.9%

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

      4. metadata-eval 99.9%

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

      5. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in v around 0 91.9%

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

      1. *-commutative 91.9%

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

      2. associate-/l* 91.9%

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

   10. Simplified 91.9%

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

   11. Taylor expanded in m around 0 60.5%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-170}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 11: 27.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.1 \cdot 10^{-36}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 5.1e-36) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.1e-36) {
		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 <= 5.1d-36) 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 <= 5.1e-36) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.1e-36:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.1e-36)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.1e-36)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.1e-36], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 5.09999999999999973e-36

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 51.9%

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

   if 5.09999999999999973e-36 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 57.3%

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

      1. distribute-rgt-in 57.3%

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

      2. *-lft-identity 57.3%

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

      3. associate--l+ 57.3%

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

      4. associate-*l/ 57.3%

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

      5. *-lft-identity 57.3%

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

      6. sub-neg 57.3%

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

      7. metadata-eval 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in m around inf 57.3%

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

      1. distribute-rgt-in 57.3%

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

      2. associate-*l/ 57.3%

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

      3. associate-*r/ 57.3%

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

      4. *-lft-identity 57.3%

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

      5. *-lft-identity 57.3%

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

   9. Simplified 57.3%

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

   10. Taylor expanded in v around inf 5.4%

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

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.1 \cdot 10^{-36}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]

Alternative 12: 27.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around inf 25.7%

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

   1. neg-mul-1 25.7%

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

   2. neg-sub0 25.7%

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

   3. associate--r- 25.7%

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

   4. metadata-eval 25.7%

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

6. Simplified 25.7%

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

7. Final simplification 25.7%

   \[\leadsto m + -1 \]

Alternative 13: 24.8% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 23.3%

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

5. Final simplification 23.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023279 
(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)))