b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.8%

Time: 7.4s

Alternatives: 14

Speedup: 1.0×

• 42.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 9.2e-19) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * ((m + -1.0) * ((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 <= 9.2d-19) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = m * ((m + (-1.0d0)) * ((m + (-1.0d0)) / v))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 9.19999999999999919e-19

   1. Initial program 100.0%

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

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

      6. distribute-rgt-neg-out 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. distribute-frac-neg 100.0%

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

      11. remove-double-neg 100.0%

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

      12. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if 9.19999999999999919e-19 < m

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. mul-1-neg 99.9%

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

      5. sub-neg 99.9%

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

   6. Simplified 99.9%

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

      1. expm1-log1p-u 97.5%

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

      2. expm1-udef 97.5%

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

   8. Applied egg-rr 97.5%

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

      1. expm1-def 97.5%

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

      2. expm1-log1p 99.9%

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

      3. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 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. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return (1.0 - m) * (((1.0 - m) * (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) * (((1.0d0 - m) * (m / v)) + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 84.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.39:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.3e-128) -1.0 (if (<= m 0.39) (/ m v) (* m (* m (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.3e-128) {
		tmp = -1.0;
	} else if (m <= 0.39) {
		tmp = m / v;
	} else {
		tmp = m * (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 <= 1.3d-128) then
        tmp = -1.0d0
    else if (m <= 0.39d0) then
        tmp = m / v
    else
        tmp = m * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.3e-128) {
		tmp = -1.0;
	} else if (m <= 0.39) {
		tmp = m / v;
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.3e-128:
		tmp = -1.0
	elif m <= 0.39:
		tmp = m / v
	else:
		tmp = m * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.3e-128)
		tmp = -1.0;
	elseif (m <= 0.39)
		tmp = Float64(m / v);
	else
		tmp = Float64(m * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.3e-128)
		tmp = -1.0;
	elseif (m <= 0.39)
		tmp = m / v;
	else
		tmp = m * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.3e-128], -1.0, If[LessEqual[m, 0.39], N[(m / v), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.2999999999999999e-128

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 65.5%

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

   if 1.2999999999999999e-128 < m < 0.39000000000000001

   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 94.1%

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

   3. Taylor expanded in v around 0 77.7%

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

   4. Taylor expanded in m around 0 77.4%

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

   if 0.39000000000000001 < 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. sub-neg 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. mul-1-neg 99.9%

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

      5. sub-neg 99.9%

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

   6. Simplified 99.9%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.9%

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

      3. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf 96.6%

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

       1. unpow2 96.6%

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

       2. associate-*r/ 96.6%

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

   13. Simplified 96.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.39:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 5: 84.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.3e-128) -1.0 (if (<= m 0.38) (/ m v) (* m (/ (* m m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.3e-128) {
		tmp = -1.0;
	} else if (m <= 0.38) {
		tmp = m / v;
	} else {
		tmp = m * ((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 <= 1.3d-128) then
        tmp = -1.0d0
    else if (m <= 0.38d0) then
        tmp = m / v
    else
        tmp = m * ((m * m) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.3e-128) {
		tmp = -1.0;
	} else if (m <= 0.38) {
		tmp = m / v;
	} else {
		tmp = m * ((m * m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.3e-128:
		tmp = -1.0
	elif m <= 0.38:
		tmp = m / v
	else:
		tmp = m * ((m * m) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.3e-128)
		tmp = -1.0;
	elseif (m <= 0.38)
		tmp = Float64(m / v);
	else
		tmp = Float64(m * Float64(Float64(m * m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.3e-128)
		tmp = -1.0;
	elseif (m <= 0.38)
		tmp = m / v;
	else
		tmp = m * ((m * m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.3e-128], -1.0, If[LessEqual[m, 0.38], N[(m / v), $MachinePrecision], N[(m * N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.2999999999999999e-128

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 65.5%

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

   if 1.2999999999999999e-128 < m < 0.38

   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 94.1%

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

   3. Taylor expanded in v around 0 77.7%

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

   4. Taylor expanded in m around 0 77.4%

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

   if 0.38 < 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. sub-neg 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. mul-1-neg 99.9%

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

      5. sub-neg 99.9%

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

   6. Simplified 99.9%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.9%

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

      3. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf 96.6%

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

       1. unpow2 96.6%

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

   13. Simplified 96.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot m}{v}\\ \end{array} \]

Alternative 6: 84.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.43:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.3e-128)
   -1.0
   (if (<= m 0.43) (* (- 1.0 m) (/ m v)) (* m (/ (* m m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.3e-128) {
		tmp = -1.0;
	} else if (m <= 0.43) {
		tmp = (1.0 - m) * (m / v);
	} else {
		tmp = m * ((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 <= 1.3d-128) then
        tmp = -1.0d0
    else if (m <= 0.43d0) then
        tmp = (1.0d0 - m) * (m / v)
    else
        tmp = m * ((m * m) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.3e-128) {
		tmp = -1.0;
	} else if (m <= 0.43) {
		tmp = (1.0 - m) * (m / v);
	} else {
		tmp = m * ((m * m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.3e-128:
		tmp = -1.0
	elif m <= 0.43:
		tmp = (1.0 - m) * (m / v)
	else:
		tmp = m * ((m * m) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.3e-128)
		tmp = -1.0;
	elseif (m <= 0.43)
		tmp = Float64(Float64(1.0 - m) * Float64(m / v));
	else
		tmp = Float64(m * Float64(Float64(m * m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.3e-128)
		tmp = -1.0;
	elseif (m <= 0.43)
		tmp = (1.0 - m) * (m / v);
	else
		tmp = m * ((m * m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.3e-128], -1.0, If[LessEqual[m, 0.43], N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.2999999999999999e-128

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 65.5%

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

   if 1.2999999999999999e-128 < m < 0.429999999999999993

   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 94.1%

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

   3. Taylor expanded in v around 0 77.7%

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

      1. frac-2neg 77.7%

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

      2. div-inv 77.5%

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

      3. distribute-rgt-neg-in 77.5%

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

      4. neg-sub0 77.5%

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

      5. metadata-eval 77.5%

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

      6. associate--r- 77.5%

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

      7. metadata-eval 77.5%

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

      8. metadata-eval 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. associate-*r/ 77.7%

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

      2. *-commutative 77.7%

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

      3. neg-mul-1 77.7%

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

      4. times-frac 77.7%

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

      5. metadata-eval 77.7%

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

      6. associate-*r/ 77.5%

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

      7. neg-mul-1 77.5%

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

      8. associate-*r/ 77.7%

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

      9. associate-*l/ 77.7%

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

      10. distribute-rgt-neg-in 77.7%

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

      11. distribute-neg-in 77.7%

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

      12. metadata-eval 77.7%

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

      13. sub-neg 77.7%

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

   7. Simplified 77.7%

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

   if 0.429999999999999993 < m

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. mul-1-neg 99.9%

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

      5. sub-neg 99.9%

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

   6. Simplified 99.9%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.9%

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

      3. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf 96.6%

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

       1. unpow2 96.6%

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

   13. Simplified 96.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.43:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot m}{v}\\ \end{array} \]

Alternative 7: 98.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 97.3%

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

      1. sub-neg 97.3%

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

      2. *-commutative 97.3%

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

      3. distribute-rgt-in 97.3%

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

      4. *-lft-identity 97.3%

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

      5. remove-double-neg 97.3%

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

      6. distribute-rgt-neg-out 97.3%

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

      7. *-commutative 97.3%

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

      8. associate-*r/ 97.4%

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

      9. *-rgt-identity 97.4%

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

      10. distribute-frac-neg 97.4%

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

      11. remove-double-neg 97.4%

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

      12. metadata-eval 97.4%

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

   6. Simplified 97.4%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around inf 96.7%

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

      1. unpow2 96.7%

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

      2. associate-*r/ 96.7%

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

      3. associate-*l* 96.7%

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

      4. neg-mul-1 96.7%

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

      5. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in v around 0 96.7%

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

      1. associate-*r/ 96.7%

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

      2. unpow2 96.7%

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

      3. *-commutative 96.7%

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

      4. associate-*r* 96.7%

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

      5. mul-1-neg 96.7%

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

   9. Simplified 96.7%

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

   10. Taylor expanded in m around 0 28.4%

       \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   11. Step-by-step derivation

       1. +-commutative 28.4%

          \[\leadsto \color{blue}{\frac{{m}^{3}}{v} + -1 \cdot \frac{{m}^{2}}{v}} \]

       2. mul-1-neg 28.4%

          \[\leadsto \frac{{m}^{3}}{v} + \color{blue}{\left(-\frac{{m}^{2}}{v}\right)} \]

       3. unpow2 28.4%

          \[\leadsto \frac{{m}^{3}}{v} + \left(-\frac{\color{blue}{m \cdot m}}{v}\right) \]

       4. unsub-neg 28.4%

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

       5. cube-mult 28.4%

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

       6. associate-/l* 28.4%

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

       7. *-lft-identity 28.4%

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

       8. associate-/l* 28.4%

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

       9. div-sub 96.7%

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

       10. sub-neg 96.7%

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

       11. metadata-eval 96.7%

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

       12. +-commutative 96.7%

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

       13. associate-/l* 96.7%

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

       14. *-commutative 96.7%

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

       15. associate-*r/ 96.7%

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

       16. associate-*l* 96.7%

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

   12. Simplified 96.7%

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

4. Final simplification 97.0%

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

Alternative 8: 98.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m / v), $MachinePrecision] * N[(m + -2.0), $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/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 97.3%

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

      1. sub-neg 97.3%

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

      2. *-commutative 97.3%

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

      3. distribute-rgt-in 97.3%

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

      4. *-lft-identity 97.3%

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

      5. remove-double-neg 97.3%

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

      6. distribute-rgt-neg-out 97.3%

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

      7. *-commutative 97.3%

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

      8. associate-*r/ 97.4%

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

      9. *-rgt-identity 97.4%

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

      10. distribute-frac-neg 97.4%

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

      11. remove-double-neg 97.4%

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

      12. metadata-eval 97.4%

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

   6. Simplified 97.4%

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

   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. sub-neg 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. mul-1-neg 99.9%

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

      5. sub-neg 99.9%

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

   6. Simplified 99.9%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.9%

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

      3. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf 52.0%

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

       1. unpow2 52.0%

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

       2. associate-*r/ 52.0%

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

       3. distribute-rgt-out 98.1%

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

   13. Simplified 98.1%

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

4. Final simplification 97.7%

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

Alternative 9: 98.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   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 97.5%

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

   if 1.6000000000000001 < 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. sub-neg 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. mul-1-neg 99.9%

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

      5. sub-neg 99.9%

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

   6. Simplified 99.9%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.9%

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

      3. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf 52.0%

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

       1. unpow2 52.0%

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

       2. associate-*r/ 52.0%

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

       3. distribute-rgt-out 98.1%

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

   13. Simplified 98.1%

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

4. Final simplification 97.8%

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

Alternative 10: 74.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4e-128) -1.0 (if (<= m 0.28) (/ m v) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4e-128) {
		tmp = -1.0;
	} else if (m <= 0.28) {
		tmp = m / v;
	} 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 <= 2.4d-128) then
        tmp = -1.0d0
    else if (m <= 0.28d0) then
        tmp = m / v
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4e-128) {
		tmp = -1.0;
	} else if (m <= 0.28) {
		tmp = m / v;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4e-128:
		tmp = -1.0
	elif m <= 0.28:
		tmp = m / v
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4e-128)
		tmp = -1.0;
	elseif (m <= 0.28)
		tmp = Float64(m / v);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4e-128)
		tmp = -1.0;
	elseif (m <= 0.28)
		tmp = m / v;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4e-128], -1.0, If[LessEqual[m, 0.28], N[(m / v), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.3999999999999998e-128

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 65.5%

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

   if 2.3999999999999998e-128 < m < 0.28000000000000003

   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 94.1%

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

   3. Taylor expanded in v around 0 77.7%

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

   4. Taylor expanded in m around 0 77.4%

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

   if 0.28000000000000003 < m

   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 0.1%

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

      1. sub-neg 0.1%

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

      2. distribute-lft-in 0.1%

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

      3. *-commutative 0.1%

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

      4. *-un-lft-identity 0.1%

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

      5. sub-neg 0.1%

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

      6. metadata-eval 0.1%

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

      7. sub-neg 0.1%

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

      8. metadata-eval 0.1%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 71.2%

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

      11. sqr-neg 71.2%

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

      12. sqrt-unprod 71.2%

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

      13. add-sqr-sqrt 71.2%

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

   4. Applied egg-rr 71.2%

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

      1. *-commutative 71.2%

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

      2. distribute-rgt1-in 71.2%

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

   6. Simplified 71.2%

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

   7. Taylor expanded in m around inf 71.2%

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

      1. unpow2 71.2%

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

      2. associate-*r/ 71.2%

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

   9. Simplified 71.2%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 11: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.38) (+ -1.0 (+ m (/ m v))) (* m (/ (* m m) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * ((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 <= 0.38d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = m * ((m * m) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = m * ((m * m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.38:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = m * ((m * m) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.38)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(m * Float64(Float64(m * m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.38)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = m * ((m * m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.38], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.38

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 97.3%

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

      1. sub-neg 97.3%

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

      2. *-commutative 97.3%

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

      3. distribute-rgt-in 97.3%

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

      4. *-lft-identity 97.3%

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

      5. remove-double-neg 97.3%

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

      6. distribute-rgt-neg-out 97.3%

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

      7. *-commutative 97.3%

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

      8. associate-*r/ 97.4%

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

      9. *-rgt-identity 97.4%

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

      10. distribute-frac-neg 97.4%

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

      11. remove-double-neg 97.4%

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

      12. metadata-eval 97.4%

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

   6. Simplified 97.4%

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

   if 0.38 < 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. sub-neg 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. mul-1-neg 99.9%

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

      5. sub-neg 99.9%

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

   6. Simplified 99.9%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.9%

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

      3. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in m around inf 96.6%

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

       1. unpow2 96.6%

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

   13. Simplified 96.6%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot m}{v}\\ \end{array} \]

Alternative 12: 62.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.4e-128) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.4e-128) {
		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 <= 1.4d-128) 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 <= 1.4e-128) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.4e-128:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.4e-128)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.4e-128)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.4e-128], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.3999999999999999e-128

   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%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 65.5%

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

   if 1.3999999999999999e-128 < m

   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 28.6%

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

   3. Taylor expanded in v around 0 23.7%

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

   4. Taylor expanded in m around 0 58.0%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 13: 26.9% 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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in v around inf 24.9%

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

   1. neg-mul-1 24.9%

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

   2. neg-sub0 24.9%

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

   3. associate--r- 24.9%

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

   4. metadata-eval 24.9%

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

6. Simplified 24.9%

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

7. Final simplification 24.9%

   \[\leadsto m + -1 \]

Alternative 14: 24.4% 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%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 22.6%

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

5. Final simplification 22.6%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023199 
(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)))