b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.4s

Alternatives: 14

Speedup: 1.0×

• 42.2% 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. Final simplification 99.9%

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

Alternative 2: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-239}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 6.8 \cdot 10^{-217}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 9.5 \cdot 10^{-209}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.25e-239)
   -1.0
   (if (<= m 6.8e-217)
     (/ m v)
     (if (<= m 9.5e-209) -1.0 (if (<= m 0.27) (/ m v) (* m (/ m v)))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.25e-239) {
		tmp = -1.0;
	} else if (m <= 6.8e-217) {
		tmp = m / v;
	} else if (m <= 9.5e-209) {
		tmp = -1.0;
	} else if (m <= 0.27) {
		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 <= 1.25d-239) then
        tmp = -1.0d0
    else if (m <= 6.8d-217) then
        tmp = m / v
    else if (m <= 9.5d-209) then
        tmp = -1.0d0
    else if (m <= 0.27d0) 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 <= 1.25e-239) {
		tmp = -1.0;
	} else if (m <= 6.8e-217) {
		tmp = m / v;
	} else if (m <= 9.5e-209) {
		tmp = -1.0;
	} else if (m <= 0.27) {
		tmp = m / v;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.25e-239:
		tmp = -1.0
	elif m <= 6.8e-217:
		tmp = m / v
	elif m <= 9.5e-209:
		tmp = -1.0
	elif m <= 0.27:
		tmp = m / v
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.25e-239)
		tmp = -1.0;
	elseif (m <= 6.8e-217)
		tmp = Float64(m / v);
	elseif (m <= 9.5e-209)
		tmp = -1.0;
	elseif (m <= 0.27)
		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 <= 1.25e-239)
		tmp = -1.0;
	elseif (m <= 6.8e-217)
		tmp = m / v;
	elseif (m <= 9.5e-209)
		tmp = -1.0;
	elseif (m <= 0.27)
		tmp = m / v;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.25e-239], -1.0, If[LessEqual[m, 6.8e-217], N[(m / v), $MachinePrecision], If[LessEqual[m, 9.5e-209], -1.0, If[LessEqual[m, 0.27], N[(m / v), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.25e-239 or 6.80000000000000032e-217 < m < 9.50000000000000028e-209

   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 89.0%

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

   if 1.25e-239 < m < 6.80000000000000032e-217 or 9.50000000000000028e-209 < m < 0.27000000000000002

   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 v around 0 69.4%

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

   5. Taylor expanded in m around 0 68.9%

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

   if 0.27000000000000002 < 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. *-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. remove-double-div 0.1%

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

      6. times-frac 0.1%

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

      7. *-un-lft-identity 0.1%

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

      8. associate-/l/ 0.1%

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

      9. associate-/r/ 0.1%

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

      10. /-rgt-identity 0.1%

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

      11. sub-neg 0.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 77.2%

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

      14. sqr-neg 77.2%

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

      15. sqrt-unprod 77.2%

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

      16. add-sqr-sqrt 77.2%

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

      17. sub-neg 77.2%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 29.2%

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

      20. sqr-neg 29.2%

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

      21. sqrt-unprod 77.2%

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

      22. add-sqr-sqrt 77.2%

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

   4. Applied egg-rr 77.2%

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

      1. *-commutative 77.2%

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

      2. distribute-rgt-in 77.2%

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

      3. +-commutative 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in m around inf 77.2%

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

      1. unpow2 77.2%

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

      2. associate-*r/ 77.2%

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

   9. Simplified 77.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-239}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 6.8 \cdot 10^{-217}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 9.5 \cdot 10^{-209}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 3: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.2 \cdot 10^{-239}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 6.2 \cdot 10^{-217}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-209}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.2e-239)
   -1.0
   (if (<= m 6.2e-217) (/ m v) (if (<= m 5e-209) -1.0 (* (/ m v) (+ m 1.0))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.2e-239) {
		tmp = -1.0;
	} else if (m <= 6.2e-217) {
		tmp = m / v;
	} else if (m <= 5e-209) {
		tmp = -1.0;
	} else {
		tmp = (m / v) * (m + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 4.2d-239) then
        tmp = -1.0d0
    else if (m <= 6.2d-217) then
        tmp = m / v
    else if (m <= 5d-209) then
        tmp = -1.0d0
    else
        tmp = (m / v) * (m + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.2e-239) {
		tmp = -1.0;
	} else if (m <= 6.2e-217) {
		tmp = m / v;
	} else if (m <= 5e-209) {
		tmp = -1.0;
	} else {
		tmp = (m / v) * (m + 1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.2e-239:
		tmp = -1.0
	elif m <= 6.2e-217:
		tmp = m / v
	elif m <= 5e-209:
		tmp = -1.0
	else:
		tmp = (m / v) * (m + 1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.2e-239)
		tmp = -1.0;
	elseif (m <= 6.2e-217)
		tmp = Float64(m / v);
	elseif (m <= 5e-209)
		tmp = -1.0;
	else
		tmp = Float64(Float64(m / v) * Float64(m + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.2e-239)
		tmp = -1.0;
	elseif (m <= 6.2e-217)
		tmp = m / v;
	elseif (m <= 5e-209)
		tmp = -1.0;
	else
		tmp = (m / v) * (m + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.2e-239], -1.0, If[LessEqual[m, 6.2e-217], N[(m / v), $MachinePrecision], If[LessEqual[m, 5e-209], -1.0, N[(N[(m / v), $MachinePrecision] * N[(m + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < 4.2000000000000004e-239 or 6.1999999999999997e-217 < m < 5.0000000000000005e-209

   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 89.0%

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

   if 4.2000000000000004e-239 < m < 6.1999999999999997e-217

   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 v around 0 75.3%

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

   5. Taylor expanded in m around 0 75.3%

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

   if 5.0000000000000005e-209 < 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 39.8%

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

      1. *-commutative 39.8%

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

      2. sub-neg 39.8%

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

      3. metadata-eval 39.8%

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

      4. distribute-lft-in 39.8%

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

      5. remove-double-div 39.8%

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

      6. times-frac 39.8%

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

      7. *-un-lft-identity 39.8%

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

      8. associate-/l/ 39.8%

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

      9. associate-/r/ 39.8%

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

      10. /-rgt-identity 39.8%

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

      11. sub-neg 39.8%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 86.0%

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

      14. sqr-neg 86.0%

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

      15. sqrt-unprod 86.0%

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

      16. add-sqr-sqrt 86.0%

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

      17. sub-neg 86.0%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 57.2%

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

      20. sqr-neg 57.2%

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

      21. sqrt-unprod 86.0%

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

      22. add-sqr-sqrt 86.0%

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

   4. Applied egg-rr 86.0%

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

      1. *-commutative 86.0%

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

      2. distribute-rgt-in 86.0%

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

      3. +-commutative 86.0%

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

   6. Simplified 86.0%

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

   7. Taylor expanded in v around 0 73.6%

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

      1. +-commutative 73.6%

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

      2. associate-*l/ 73.6%

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

      3. +-commutative 73.6%

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

   9. Simplified 73.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.2 \cdot 10^{-239}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 6.2 \cdot 10^{-217}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-209}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + 1\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 10^{-27}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 + m \cdot \left(m + -2\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1e-27) (+ -1.0 (+ m (/ m v))) (/ (* m (+ 1.0 (* m (+ m -2.0)))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1e-27) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (1.0 + (m * (m + -2.0)))) / 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 <= 1d-27) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m * (1.0d0 + (m * (m + (-2.0d0))))) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1e-27) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (1.0 + (m * (m + -2.0)))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1e-27:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m * (1.0 + (m * (m + -2.0)))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1e-27)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(1.0 + Float64(m * Float64(m + -2.0)))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1e-27)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m * (1.0 + (m * (m + -2.0)))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1e-27], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(1.0 + N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1e-27

   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 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-in 99.7%

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

      6. *-lft-identity 99.7%

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

      7. associate-*l/ 100.0%

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

      8. *-lft-identity 100.0%

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

   6. Simplified 100.0%

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

   if 1e-27 < 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 v around 0 99.6%

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

   5. Taylor expanded in m around 0 99.6%

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

      1. unpow2 99.6%

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

      2. distribute-rgt-out 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 10^{-27}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 + m \cdot \left(m + -2\right)\right)}{v}\\ \end{array} \]

Alternative 5: 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 6: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.35d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m * (m / v)) * (m + (-1.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 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/ 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 99.0%

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

      1. sub-neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. +-commutative 99.0%

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

      4. *-commutative 99.0%

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

      5. distribute-rgt-in 99.0%

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

      6. *-lft-identity 99.0%

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

      7. associate-*l/ 99.3%

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

      8. *-lft-identity 99.3%

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

   6. Simplified 99.3%

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

   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. 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 97.4%

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

      1. associate-*r/ 97.4%

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

      2. mul-1-neg 97.4%

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

      3. unpow2 97.4%

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

      4. distribute-rgt-neg-out 97.4%

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

      5. associate-*l/ 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in v around 0 97.4%

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

      1. mul-1-neg 97.4%

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

      2. unpow2 97.4%

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

      3. associate-/l* 97.4%

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

      4. distribute-neg-frac 97.4%

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

   9. Simplified 97.4%

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

   10. Taylor expanded in m around 0 22.6%

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

       1. *-commutative 22.6%

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

       2. unpow2 22.6%

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

       3. associate-*r/ 22.6%

          \[\leadsto \color{blue}{\left(m \cdot \frac{m}{v}\right)} \cdot -1 + \frac{{m}^{3}}{v} \]

       4. unpow3 22.6%

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

       5. associate-*l/ 22.6%

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

       6. associate-*r/ 22.6%

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

       7. distribute-lft-out 97.4%

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

   12. Simplified 97.4%

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

4. Final simplification 98.3%

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

Alternative 7: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 99.3%

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

   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. 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 97.4%

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

      1. associate-*r/ 97.4%

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

      2. mul-1-neg 97.4%

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

      3. unpow2 97.4%

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

      4. distribute-rgt-neg-out 97.4%

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

      5. associate-*l/ 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in v around 0 97.4%

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

      1. mul-1-neg 97.4%

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

      2. unpow2 97.4%

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

      3. associate-/l* 97.4%

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

      4. distribute-neg-frac 97.4%

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

   9. Simplified 97.4%

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

   10. Taylor expanded in m around 0 22.6%

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

       1. *-commutative 22.6%

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

       2. unpow2 22.6%

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

       3. associate-*r/ 22.6%

          \[\leadsto \color{blue}{\left(m \cdot \frac{m}{v}\right)} \cdot -1 + \frac{{m}^{3}}{v} \]

       4. unpow3 22.6%

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

       5. associate-*l/ 22.6%

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

       6. associate-*r/ 22.6%

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

       7. distribute-lft-out 97.4%

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

   12. Simplified 97.4%

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

4. Final simplification 98.3%

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

Alternative 8: 98.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.62) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m * (m * (m + -2.0))) / 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 <= 1.62d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = (m * (m * (m + (-2.0d0)))) / v
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1.6200000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 99.3%

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

   if 1.6200000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      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 0 99.9%

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

   5. Taylor expanded in m around inf 98.8%

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

      1. unpow2 98.8%

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

      2. distribute-rgt-out 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 99.1%

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

Alternative 9: 98.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1.6200000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 99.3%

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

   if 1.6200000000000001 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      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 0 99.9%

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

   5. Taylor expanded in m around inf 50.8%

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

      1. unpow2 50.8%

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

      2. cube-mult 50.7%

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

      3. distribute-rgt-out 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 99.1%

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

Alternative 10: 87.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.35d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m / v) * (m + 1.0d0)
    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 + (m / v));
	} else {
		tmp = (m / v) * (m + 1.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 2.35], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m + 1.0), $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/ 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 99.0%

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

      1. sub-neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. +-commutative 99.0%

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

      4. *-commutative 99.0%

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

      5. distribute-rgt-in 99.0%

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

      6. *-lft-identity 99.0%

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

      7. associate-*l/ 99.3%

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

      8. *-lft-identity 99.3%

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

   6. Simplified 99.3%

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

   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. 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. *-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. remove-double-div 0.1%

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

      6. times-frac 0.1%

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

      7. *-un-lft-identity 0.1%

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

      8. associate-/l/ 0.1%

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

      9. associate-/r/ 0.1%

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

      10. /-rgt-identity 0.1%

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

      11. sub-neg 0.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 77.2%

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

      14. sqr-neg 77.2%

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

      15. sqrt-unprod 77.2%

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

      16. add-sqr-sqrt 77.2%

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

      17. sub-neg 77.2%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 29.2%

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

      20. sqr-neg 29.2%

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

      21. sqrt-unprod 77.2%

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

      22. add-sqr-sqrt 77.2%

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

   4. Applied egg-rr 77.2%

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

      1. *-commutative 77.2%

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

      2. distribute-rgt-in 77.2%

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

      3. +-commutative 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in v around 0 77.2%

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

      1. +-commutative 77.2%

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

      2. associate-*l/ 77.2%

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

      3. +-commutative 77.2%

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

   9. Simplified 77.2%

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

4. Final simplification 88.0%

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

Alternative 11: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.375:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.375) (+ -1.0 (+ m (/ m v))) (/ (* m (* m m)) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.375) {
		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 <= 0.375d0) 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 <= 0.375) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * (m * m)) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.375:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m * (m * m)) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.375)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(m * m)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.375)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m * (m * m)) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.375], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.375

   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 99.0%

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

      1. sub-neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. +-commutative 99.0%

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

      4. *-commutative 99.0%

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

      5. distribute-rgt-in 99.0%

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

      6. *-lft-identity 99.0%

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

      7. associate-*l/ 99.3%

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

      8. *-lft-identity 99.3%

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

   6. Simplified 99.3%

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

   if 0.375 < 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 v around 0 99.9%

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

   5. Taylor expanded in m around inf 97.3%

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

      1. unpow2 97.3%

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

   7. Simplified 97.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.375:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

Alternative 12: 59.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 1.6e-81) (/ m v) (+ m -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 1.6e-81) {
		tmp = m / v;
	} else {
		tmp = m + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 1.6d-81) then
        tmp = m / v
    else
        tmp = m + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 1.6e-81) {
		tmp = m / v;
	} else {
		tmp = m + -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 1.6e-81:
		tmp = m / v
	else:
		tmp = m + -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 1.6e-81)
		tmp = Float64(m / v);
	else
		tmp = Float64(m + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 1.6e-81)
		tmp = m / v;
	else
		tmp = m + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 1.6e-81], N[(m / v), $MachinePrecision], N[(m + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.6e-81

   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/ 99.9%

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

      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 0 84.9%

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

   5. Taylor expanded in m around 0 62.8%

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

   if 1.6e-81 < v

   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 44.4%

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

      1. neg-mul-1 44.4%

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

      2. neg-sub0 44.4%

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

      3. associate--r- 44.4%

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

      4. metadata-eval 44.4%

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

   6. Simplified 44.4%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 13: 27.0% 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 21.4%

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

   1. neg-mul-1 21.4%

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

   2. neg-sub0 21.4%

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

   3. associate--r- 21.4%

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

   4. metadata-eval 21.4%

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

6. Simplified 21.4%

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

7. Final simplification 21.4%

   \[\leadsto m + -1 \]

Alternative 14: 24.5% 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 18.9%

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

5. Final simplification 18.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023214 
(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)))