b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.2s

Alternatives: 15

Speedup: 1.0×

• 41.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 97.1%

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

      1. expm1-log1p-u 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. expm1-undefine 97.1%

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

      2. sub-neg 97.1%

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

      3. log1p-undefine 97.1%

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

      4. rem-exp-log 97.1%

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

      5. associate-+r- 97.1%

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

      6. metadata-eval 97.1%

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

      7. metadata-eval 97.1%

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

   7. Simplified 97.1%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.9%

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

      4. associate-*l/ 99.9%

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

      5. *-un-lft-identity 99.9%

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

      6. associate-/r* 99.9%

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

      7. neg-mul-1 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around inf 99.9%

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

   8. Taylor expanded in m around inf 97.2%

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

      1. neg-mul-1 97.2%

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

   10. Simplified 97.2%

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

   11. Taylor expanded in v around 0 97.3%

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

4. Final simplification 97.2%

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

Alternative 3: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (-1.0 + (m / v)) * (-1.0 - (m - 2.0));
	} else {
		tmp = m * (((m * (m + -1.0)) / v) - -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 / v)) * ((-1.0d0) - (m - 2.0d0))
    else
        tmp = m * (((m * (m + (-1.0d0))) / v) - (-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 / v)) * (-1.0 - (m - 2.0));
	} else {
		tmp = m * (((m * (m + -1.0)) / v) - -1.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(m - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(N[(m * N[(m + -1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - -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 97.1%

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

      1. expm1-log1p-u 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. expm1-undefine 97.1%

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

      2. sub-neg 97.1%

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

      3. log1p-undefine 97.1%

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

      4. rem-exp-log 97.1%

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

      5. associate-+r- 97.1%

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

      6. metadata-eval 97.1%

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

      7. metadata-eval 97.1%

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

   7. Simplified 97.1%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 97.3%

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

      1. neg-mul-1 97.2%

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

   5. Simplified 97.3%

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

4. Final simplification 97.2%

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

Alternative 4: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 97.1%

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

      1. expm1-log1p-u 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. expm1-undefine 97.1%

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

      2. sub-neg 97.1%

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

      3. log1p-undefine 97.1%

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

      4. rem-exp-log 97.1%

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

      5. associate-+r- 97.1%

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

      6. metadata-eval 97.1%

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

      7. metadata-eval 97.1%

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

   7. Simplified 97.1%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around inf 97.2%

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

      1. neg-mul-1 97.2%

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

   7. Simplified 97.2%

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

4. Final simplification 97.2%

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

Alternative 5: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\left(-1 + \frac{m}{v}\right) \cdot \left(-1 - \left(m - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.43)
   (* (+ -1.0 (/ m v)) (- -1.0 (- m 2.0)))
   (* m (+ 1.0 (/ (* m m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = (-1.0 + (m / v)) * (-1.0 - (m - 2.0));
	} else {
		tmp = m * (1.0 + ((m * m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 0.43d0) then
        tmp = ((-1.0d0) + (m / v)) * ((-1.0d0) - (m - 2.0d0))
    else
        tmp = m * (1.0d0 + ((m * m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = (-1.0 + (m / v)) * (-1.0 - (m - 2.0));
	} else {
		tmp = m * (1.0 + ((m * m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.43:
		tmp = (-1.0 + (m / v)) * (-1.0 - (m - 2.0))
	else:
		tmp = m * (1.0 + ((m * m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.43)
		tmp = Float64(Float64(-1.0 + Float64(m / v)) * Float64(-1.0 - Float64(m - 2.0)));
	else
		tmp = Float64(m * Float64(1.0 + Float64(Float64(m * m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.43)
		tmp = (-1.0 + (m / v)) * (-1.0 - (m - 2.0));
	else
		tmp = m * (1.0 + ((m * m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.43], N[(N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(m - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(1.0 + N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.429999999999999993

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 97.1%

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

      1. expm1-log1p-u 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. expm1-undefine 97.1%

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

      2. sub-neg 97.1%

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

      3. log1p-undefine 97.1%

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

      4. rem-exp-log 97.1%

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

      5. associate-+r- 97.1%

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

      6. metadata-eval 97.1%

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

      7. metadata-eval 97.1%

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

   7. Simplified 97.1%

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

   if 0.429999999999999993 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 97.3%

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

      1. neg-mul-1 97.2%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in m around inf 97.2%

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

      1. neg-mul-1 97.2%

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

   8. Simplified 97.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\left(-1 + \frac{m}{v}\right) \cdot \left(-1 - \left(m - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m \cdot m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (-1.0 + (m / v)) * (-1.0 - (m - 2.0));
	} 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 <= 1.0d0) then
        tmp = ((-1.0d0) + (m / v)) * ((-1.0d0) - (m - 2.0d0))
    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 <= 1.0) {
		tmp = (-1.0 + (m / v)) * (-1.0 - (m - 2.0));
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(-1.0 + Float64(m / v)) * Float64(-1.0 - Float64(m - 2.0)));
	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 <= 1.0)
		tmp = (-1.0 + (m / v)) * (-1.0 - (m - 2.0));
	else
		tmp = m * ((m + 1.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

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

   3. Taylor expanded in m around 0 97.1%

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

      1. expm1-log1p-u 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. expm1-undefine 97.1%

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

      2. sub-neg 97.1%

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

      3. log1p-undefine 97.1%

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

      4. rem-exp-log 97.1%

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

      5. associate-+r- 97.1%

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

      6. metadata-eval 97.1%

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

      7. metadata-eval 97.1%

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

   7. Simplified 97.1%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-lft-in 0.1%

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

      5. fma-define 0.1%

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

      6. sub-neg 0.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 75.9%

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

      9. sqr-neg 75.9%

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

      10. sqrt-unprod 75.9%

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

      11. add-sqr-sqrt 75.9%

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

      12. *-commutative 75.9%

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

      13. neg-mul-1 75.9%

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

      14. add-sqr-sqrt 75.9%

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

      15. sqrt-unprod 75.9%

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

      16. sqr-neg 75.9%

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

      17. sqrt-unprod 0.0%

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

      18. add-sqr-sqrt 75.9%

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

      19. sub-neg 75.9%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 75.9%

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

      22. sqr-neg 75.9%

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

      23. sqrt-unprod 75.9%

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

      24. add-sqr-sqrt 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. fma-undefine 75.9%

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

      2. *-rgt-identity 75.9%

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

      3. distribute-lft-in 75.9%

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

      4. +-commutative 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in v around 0 75.9%

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

      1. associate-*r/ 75.9%

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

      2. +-commutative 75.9%

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

   10. Simplified 75.9%

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

4. Final simplification 86.8%

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

Alternative 7: 87.3% accurate, 0.9× speedup

Math

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

FPCore

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

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);
	}
	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)
    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);
	}
	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)
	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 + 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 + 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[(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. Add Preprocessing

   3. Taylor expanded in m around 0 97.1%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-lft-in 0.1%

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

      5. fma-define 0.1%

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

      6. sub-neg 0.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 75.9%

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

      9. sqr-neg 75.9%

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

      10. sqrt-unprod 75.9%

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

      11. add-sqr-sqrt 75.9%

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

      12. *-commutative 75.9%

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

      13. neg-mul-1 75.9%

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

      14. add-sqr-sqrt 75.9%

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

      15. sqrt-unprod 75.9%

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

      16. sqr-neg 75.9%

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

      17. sqrt-unprod 0.0%

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

      18. add-sqr-sqrt 75.9%

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

      19. sub-neg 75.9%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 75.9%

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

      22. sqr-neg 75.9%

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

      23. sqrt-unprod 75.9%

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

      24. add-sqr-sqrt 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. fma-undefine 75.9%

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

      2. *-rgt-identity 75.9%

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

      3. distribute-lft-in 75.9%

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

      4. +-commutative 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in v around 0 75.9%

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

      1. associate-*r/ 75.9%

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

      2. +-commutative 75.9%

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

   10. Simplified 75.9%

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

4. Final simplification 86.8%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. distribute-rgt-in 76.8%

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

   3. associate-/r/ 76.8%

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

   4. unpow-1 76.8%

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

   5. neg-mul-1 76.8%

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

   6. distribute-lft-neg-out 76.8%

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

   7. associate-/r/ 76.8%

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

   8. sub-neg 76.8%

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

   9. unpow-1 76.8%

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

   10. div-sub 99.8%

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

7. Simplified 99.8%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 10: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.35:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.35) (+ -1.0 (/ m v)) (* m (/ (+ m 1.0) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.35) {
		tmp = -1.0 + (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 <= 2.35d0) then
        tmp = (-1.0d0) + (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 <= 2.35) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * ((m + 1.0) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.35:
		tmp = -1.0 + (m / v)
	else:
		tmp = m * ((m + 1.0) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.35)
		tmp = Float64(-1.0 + 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 <= 2.35)
		tmp = -1.0 + (m / v);
	else
		tmp = m * ((m + 1.0) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.35], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m + 1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.35000000000000009

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-rgt-in 99.7%

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

      2. associate-*r/ 100.0%

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

      3. clear-num 99.8%

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

      4. associate-*l/ 99.8%

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

      5. *-un-lft-identity 99.8%

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

      6. associate-/r* 99.8%

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

      7. neg-mul-1 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in m around 0 97.0%

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

   8. Taylor expanded in m around 0 97.0%

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

   if 2.35000000000000009 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-lft-in 0.1%

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

      5. fma-define 0.1%

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

      6. sub-neg 0.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 75.9%

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

      9. sqr-neg 75.9%

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

      10. sqrt-unprod 75.9%

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

      11. add-sqr-sqrt 75.9%

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

      12. *-commutative 75.9%

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

      13. neg-mul-1 75.9%

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

      14. add-sqr-sqrt 75.9%

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

      15. sqrt-unprod 75.9%

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

      16. sqr-neg 75.9%

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

      17. sqrt-unprod 0.0%

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

      18. add-sqr-sqrt 75.9%

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

      19. sub-neg 75.9%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 75.9%

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

      22. sqr-neg 75.9%

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

      23. sqrt-unprod 75.9%

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

      24. add-sqr-sqrt 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. fma-undefine 75.9%

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

      2. *-rgt-identity 75.9%

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

      3. distribute-lft-in 75.9%

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

      4. +-commutative 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in v around 0 75.9%

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

      1. associate-*r/ 75.9%

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

      2. +-commutative 75.9%

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

   10. Simplified 75.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.35:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m + 1}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.3 \cdot 10^{-44}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3.3e-44) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.3e-44) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 3.3d-44) then
        tmp = -1.0d0
    else
        tmp = m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.3e-44) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.3e-44:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.3e-44)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.3e-44)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.3e-44], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 3.30000000000000006e-44

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 60.5%

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

   if 3.30000000000000006e-44 < 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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around inf 5.2%

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

      1. neg-mul-1 5.2%

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

      2. sub-neg 5.2%

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

      3. +-commutative 5.2%

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

      4. distribute-neg-in 5.2%

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

      5. remove-double-neg 5.2%

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

      6. metadata-eval 5.2%

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

   7. Simplified 5.2%

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

   8. Taylor expanded in m around inf 5.7%

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

Alternative 12: 25.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3 \cdot 10^{-44}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3e-44) -1.0 1.0))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3e-44) {
		tmp = -1.0;
	} else {
		tmp = 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 <= 3d-44) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3e-44) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3e-44:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3e-44)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3e-44)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3e-44], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if m < 3.0000000000000002e-44

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 60.5%

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

   if 3.0000000000000002e-44 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 10.1%

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

      1. *-commutative 10.1%

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

      2. sub-neg 10.1%

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

      3. metadata-eval 10.1%

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

      4. distribute-lft-in 10.1%

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

      5. fma-define 10.1%

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

      6. sub-neg 10.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 76.1%

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

      9. sqr-neg 76.1%

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

      10. sqrt-unprod 76.1%

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

      11. add-sqr-sqrt 76.1%

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

      12. *-commutative 76.1%

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

      13. neg-mul-1 76.1%

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

      14. add-sqr-sqrt 66.2%

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

      15. sqrt-unprod 76.1%

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

      16. sqr-neg 76.1%

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

      17. sqrt-unprod 9.8%

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

      18. add-sqr-sqrt 76.1%

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

      19. sub-neg 76.1%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 76.1%

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

      22. sqr-neg 76.1%

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

      23. sqrt-unprod 76.1%

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

      24. add-sqr-sqrt 76.1%

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

   5. Applied egg-rr 76.1%

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

      1. fma-undefine 76.1%

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

      2. *-rgt-identity 76.1%

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

      3. distribute-lft-in 76.1%

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

      4. +-commutative 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in m around 0 3.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 75.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. distribute-rgt-in 99.8%

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

   2. associate-*r/ 99.9%

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

   3. clear-num 99.8%

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

   4. associate-*l/ 99.9%

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

   5. *-un-lft-identity 99.9%

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

   6. associate-/r* 99.8%

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

   7. neg-mul-1 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in m around 0 75.0%

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

8. Taylor expanded in m around 0 75.0%

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

9. Final simplification 75.0%

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

Alternative 14: 26.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 29.8%

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

   1. neg-mul-1 29.8%

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

   2. sub-neg 29.8%

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

   3. +-commutative 29.8%

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

   4. distribute-neg-in 29.8%

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

   5. remove-double-neg 29.8%

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

   6. metadata-eval 29.8%

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

7. Simplified 29.8%

   \[\leadsto \color{blue}{m + -1} \]
8. Add Preprocessing

Alternative 15: 24.3% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 27.4%

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

Reproduce

herbie shell --seed 2024121 
(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)))