b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 9.7s

Alternatives: 11

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.8%

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

   2. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.9 \cdot 10^{-142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.44 \cdot 10^{-92}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.9e-142)
   -1.0
   (if (<= m 3.3e-104)
     (/ m v)
     (if (<= m 1.44e-92) -1.0 (if (<= m 0.28) (/ m v) (* m (/ m v)))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.9e-142) {
		tmp = -1.0;
	} else if (m <= 3.3e-104) {
		tmp = m / v;
	} else if (m <= 1.44e-92) {
		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 <= 1.9d-142) then
        tmp = -1.0d0
    else if (m <= 3.3d-104) then
        tmp = m / v
    else if (m <= 1.44d-92) 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 <= 1.9e-142) {
		tmp = -1.0;
	} else if (m <= 3.3e-104) {
		tmp = m / v;
	} else if (m <= 1.44e-92) {
		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 <= 1.9e-142:
		tmp = -1.0
	elif m <= 3.3e-104:
		tmp = m / v
	elif m <= 1.44e-92:
		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 <= 1.9e-142)
		tmp = -1.0;
	elseif (m <= 3.3e-104)
		tmp = Float64(m / v);
	elseif (m <= 1.44e-92)
		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 <= 1.9e-142)
		tmp = -1.0;
	elseif (m <= 3.3e-104)
		tmp = m / v;
	elseif (m <= 1.44e-92)
		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, 1.9e-142], -1.0, If[LessEqual[m, 3.3e-104], N[(m / v), $MachinePrecision], If[LessEqual[m, 1.44e-92], -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 < 1.89999999999999986e-142 or 3.30000000000000002e-104 < m < 1.4400000000000001e-92

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 76.3%

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

   if 1.89999999999999986e-142 < m < 3.30000000000000002e-104 or 1.4400000000000001e-92 < m < 0.28000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 94.8%

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

   4. Taylor expanded in v around 0 94.8%

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

      1. +-commutative 94.8%

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

      2. sub-neg 94.8%

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

      3. metadata-eval 94.8%

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

      4. distribute-neg-in 94.8%

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

      5. associate-*r* 94.8%

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

      6. sub-neg 94.8%

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

      7. metadata-eval 94.8%

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

      8. distribute-neg-in 94.8%

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

      9. distribute-rgt-out 94.8%

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

      10. distribute-neg-in 94.8%

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

      11. metadata-eval 94.8%

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

      12. sub-neg 94.8%

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

      13. neg-mul-1 94.8%

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

   6. Simplified 94.8%

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

   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. associate-/l* 76.8%

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

   9. Simplified 76.8%

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

   10. Taylor expanded in m around 0 77.0%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.3%

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

      1. *-commutative 0.3%

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

      2. sub-neg 0.3%

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

      3. metadata-eval 0.3%

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

      4. distribute-rgt-in 0.3%

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

      5. *-commutative 0.3%

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

      6. add-sqr-sqrt 0.2%

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

      7. sqrt-unprod 77.4%

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

      8. sqr-neg 77.4%

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

      9. sqrt-unprod 77.3%

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

      10. add-sqr-sqrt 77.3%

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

      11. neg-sub0 77.3%

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

      12. associate--r- 77.3%

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

      13. metadata-eval 77.3%

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

      14. neg-mul-1 77.3%

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

      15. neg-sub0 77.3%

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

      16. associate--r- 77.3%

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

      17. metadata-eval 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. *-commutative 77.3%

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

      2. distribute-lft1-in 77.3%

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

      3. +-commutative 77.3%

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

   7. Simplified 77.3%

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

   8. Taylor expanded in v around 0 77.3%

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

      1. sub-neg 77.3%

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

      2. metadata-eval 77.3%

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

      3. +-commutative 77.3%

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

      4. associate-*r/ 77.3%

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

   10. Simplified 77.3%

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

   11. Taylor expanded in m around inf 77.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.9 \cdot 10^{-142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.44 \cdot 10^{-92}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-144}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 3.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.44 \cdot 10^{-92}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 + m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7e-144)
   -1.0
   (if (<= m 3.2e-104)
     (/ m v)
     (if (<= m 1.44e-92) -1.0 (* m (/ (+ 1.0 m) v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7e-144) {
		tmp = -1.0;
	} else if (m <= 3.2e-104) {
		tmp = m / v;
	} else if (m <= 1.44e-92) {
		tmp = -1.0;
	} else {
		tmp = m * ((1.0 + m) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 7d-144) then
        tmp = -1.0d0
    else if (m <= 3.2d-104) then
        tmp = m / v
    else if (m <= 1.44d-92) then
        tmp = -1.0d0
    else
        tmp = m * ((1.0d0 + m) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7e-144) {
		tmp = -1.0;
	} else if (m <= 3.2e-104) {
		tmp = m / v;
	} else if (m <= 1.44e-92) {
		tmp = -1.0;
	} else {
		tmp = m * ((1.0 + m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7e-144:
		tmp = -1.0
	elif m <= 3.2e-104:
		tmp = m / v
	elif m <= 1.44e-92:
		tmp = -1.0
	else:
		tmp = m * ((1.0 + m) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7e-144)
		tmp = -1.0;
	elseif (m <= 3.2e-104)
		tmp = Float64(m / v);
	elseif (m <= 1.44e-92)
		tmp = -1.0;
	else
		tmp = Float64(m * Float64(Float64(1.0 + m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7e-144)
		tmp = -1.0;
	elseif (m <= 3.2e-104)
		tmp = m / v;
	elseif (m <= 1.44e-92)
		tmp = -1.0;
	else
		tmp = m * ((1.0 + m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7e-144], -1.0, If[LessEqual[m, 3.2e-104], N[(m / v), $MachinePrecision], If[LessEqual[m, 1.44e-92], -1.0, N[(m * N[(N[(1.0 + m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < 6.9999999999999997e-144 or 3.19999999999999989e-104 < m < 1.4400000000000001e-92

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 76.3%

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

   if 6.9999999999999997e-144 < m < 3.19999999999999989e-104

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 100.0%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. associate-*r* 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. distribute-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 80.2%

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

      1. associate-/l* 79.9%

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

   9. Simplified 79.9%

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

   10. Taylor expanded in m around 0 80.2%

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

   if 1.4400000000000001e-92 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 22.6%

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

      1. *-commutative 22.6%

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

      2. sub-neg 22.6%

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

      3. metadata-eval 22.6%

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

      4. distribute-rgt-in 22.6%

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

      5. *-commutative 22.6%

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

      6. add-sqr-sqrt 22.5%

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

      7. sqrt-unprod 81.1%

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

      8. sqr-neg 81.1%

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

      9. sqrt-unprod 58.7%

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

      10. add-sqr-sqrt 62.8%

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

      11. neg-sub0 62.8%

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

      12. associate--r- 62.8%

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

      13. metadata-eval 62.8%

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

      14. neg-mul-1 62.8%

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

      15. neg-sub0 62.8%

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

      16. associate--r- 62.8%

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

      17. metadata-eval 62.8%

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

   5. Applied egg-rr 62.8%

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

      1. *-commutative 62.8%

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

      2. distribute-lft1-in 62.8%

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

      3. +-commutative 62.8%

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

   7. Simplified 62.8%

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

   8. Taylor expanded in v around 0 59.2%

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

      1. sub-neg 59.2%

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

      2. metadata-eval 59.2%

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

      3. +-commutative 59.2%

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

      4. associate-*r/ 59.2%

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

   10. Simplified 59.2%

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

   12. Applied egg-rr 76.9%

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

       1. /-rgt-identity 76.9%

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

       2. associate-*r/ 76.9%

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

       3. hypot-undefine 76.9%

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

       4. hypot-undefine 76.9%

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

       5. rem-square-sqrt 76.9%

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

       6. metadata-eval 76.9%

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

       7. rem-square-sqrt 76.9%

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

       8. *-rgt-identity 76.9%

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

       9. distribute-rgt1-in 76.9%

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

       10. distribute-lft1-in 76.9%

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

       11. *-rgt-identity 76.9%

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

   14. Simplified 76.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-144}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 3.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.44 \cdot 10^{-92}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 + m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.3 \cdot 10^{-142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{-104} \lor \neg \left(m \leq 1.44 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.3e-142)
   -1.0
   (if (or (<= m 3.3e-104) (not (<= m 1.44e-92))) (/ m v) -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.3e-142) {
		tmp = -1.0;
	} else if ((m <= 3.3e-104) || !(m <= 1.44e-92)) {
		tmp = m / v;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 3.3d-142) then
        tmp = -1.0d0
    else if ((m <= 3.3d-104) .or. (.not. (m <= 1.44d-92))) then
        tmp = m / v
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.3e-142) {
		tmp = -1.0;
	} else if ((m <= 3.3e-104) || !(m <= 1.44e-92)) {
		tmp = m / v;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.3e-142:
		tmp = -1.0
	elif (m <= 3.3e-104) or not (m <= 1.44e-92):
		tmp = m / v
	else:
		tmp = -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.3e-142)
		tmp = -1.0;
	elseif ((m <= 3.3e-104) || !(m <= 1.44e-92))
		tmp = Float64(m / v);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.3e-142)
		tmp = -1.0;
	elseif ((m <= 3.3e-104) || ~((m <= 1.44e-92)))
		tmp = m / v;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.3e-142], -1.0, If[Or[LessEqual[m, 3.3e-104], N[Not[LessEqual[m, 1.44e-92]], $MachinePrecision]], N[(m / v), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if m < 3.2999999999999997e-142 or 3.30000000000000002e-104 < m < 1.4400000000000001e-92

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 76.3%

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

   if 3.2999999999999997e-142 < m < 3.30000000000000002e-104 or 1.4400000000000001e-92 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 29.2%

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

   4. Taylor expanded in v around 0 29.2%

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

      1. +-commutative 29.2%

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

      2. sub-neg 29.2%

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

      3. metadata-eval 29.2%

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

      4. distribute-neg-in 29.2%

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

      5. associate-*r* 29.2%

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

      6. sub-neg 29.2%

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

      7. metadata-eval 29.2%

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

      8. distribute-neg-in 29.2%

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

      9. distribute-rgt-out 29.2%

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

      10. distribute-neg-in 29.2%

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

      11. metadata-eval 29.2%

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

      12. sub-neg 29.2%

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

      13. neg-mul-1 29.2%

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

   6. Simplified 29.2%

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

   7. Taylor expanded in v around 0 23.8%

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

      1. associate-/l* 23.7%

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

   9. Simplified 23.7%

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

   10. Taylor expanded in m around 0 64.2%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.3 \cdot 10^{-142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{-104} \lor \neg \left(m \leq 1.44 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 97.0%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around inf 97.8%

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

      1. neg-mul-1 97.8%

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

      2. distribute-neg-frac2 97.8%

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

   7. Simplified 97.8%

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

4. Final simplification 97.4%

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

Alternative 6: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 97.0%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around inf 97.8%

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

      1. neg-mul-1 97.8%

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

      2. distribute-neg-frac2 97.8%

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

   7. Simplified 97.8%

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

      1. *-commutative 97.8%

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

      2. distribute-frac-neg2 97.8%

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

      3. distribute-frac-neg 97.8%

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

      4. associate-*l/ 97.9%

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

   9. Applied egg-rr 97.9%

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

4. Final simplification 97.5%

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

Alternative 7: 87.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 97.0%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. sub-neg 0.1%

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

      3. metadata-eval 0.1%

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

      4. distribute-rgt-in 0.1%

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

      5. *-commutative 0.1%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 77.9%

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

      8. sqr-neg 77.9%

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

      9. sqrt-unprod 77.9%

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

      10. add-sqr-sqrt 77.9%

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

      11. neg-sub0 77.9%

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

      12. associate--r- 77.9%

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

      13. metadata-eval 77.9%

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

      14. neg-mul-1 77.9%

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

      15. neg-sub0 77.9%

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

      16. associate--r- 77.9%

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

      17. metadata-eval 77.9%

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

   5. Applied egg-rr 77.9%

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

      1. *-commutative 77.9%

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

      2. distribute-lft1-in 77.9%

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

      3. +-commutative 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in v around 0 77.9%

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

      1. sub-neg 77.9%

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

      2. metadata-eval 77.9%

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

      3. +-commutative 77.9%

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

      4. associate-*r/ 77.9%

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

   10. Simplified 77.9%

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

   12. Applied egg-rr 77.9%

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

       1. /-rgt-identity 77.9%

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

       2. associate-*r/ 77.9%

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

       3. hypot-undefine 77.9%

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

       4. hypot-undefine 77.9%

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

       5. rem-square-sqrt 77.9%

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

       6. metadata-eval 77.9%

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

       7. rem-square-sqrt 77.9%

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

       8. *-rgt-identity 77.9%

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

       9. distribute-rgt1-in 77.9%

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

       10. distribute-lft1-in 77.9%

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

       11. *-rgt-identity 77.9%

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

   14. Simplified 77.9%

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

4. Final simplification 87.5%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(m + -1.0), $MachinePrecision] * N[(N[(m * N[(N[(m + -1.0), $MachinePrecision] / 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* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 87.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.28) {
		tmp = (m / v) + (m + -1.0);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 0.28d0) then
        tmp = (m / v) + (m + (-1.0d0))
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.28000000000000003

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. fma-define 99.7%

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

      3. *-commutative 99.7%

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

      4. neg-mul-1 99.7%

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

      5. fmm-undef 99.7%

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

      6. associate-*r/ 100.0%

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

      7. clear-num 99.7%

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

      8. un-div-inv 99.7%

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

      9. associate-/r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in m around 0 97.6%

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

   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. Add Preprocessing

   3. Taylor expanded in m around 0 0.3%

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

      1. *-commutative 0.3%

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

      2. sub-neg 0.3%

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

      3. metadata-eval 0.3%

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

      4. distribute-rgt-in 0.3%

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

      5. *-commutative 0.3%

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

      6. add-sqr-sqrt 0.2%

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

      7. sqrt-unprod 77.4%

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

      8. sqr-neg 77.4%

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

      9. sqrt-unprod 77.3%

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

      10. add-sqr-sqrt 77.3%

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

      11. neg-sub0 77.3%

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

      12. associate--r- 77.3%

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

      13. metadata-eval 77.3%

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

      14. neg-mul-1 77.3%

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

      15. neg-sub0 77.3%

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

      16. associate--r- 77.3%

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

      17. metadata-eval 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. *-commutative 77.3%

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

      2. distribute-lft1-in 77.3%

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

      3. +-commutative 77.3%

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

   7. Simplified 77.3%

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

   8. Taylor expanded in v around 0 77.3%

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

      1. sub-neg 77.3%

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

      2. metadata-eval 77.3%

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

      3. +-commutative 77.3%

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

      4. associate-*r/ 77.3%

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

   10. Simplified 77.3%

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

   11. Taylor expanded in m around inf 77.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 27.8%

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

   1. neg-mul-1 27.8%

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

   2. sub-neg 27.8%

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

   3. +-commutative 27.8%

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

   4. distribute-neg-in 27.8%

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

   5. remove-double-neg 27.8%

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

   6. metadata-eval 27.8%

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

7. Simplified 27.8%

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

8. Final simplification 27.8%

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

Alternative 11: 24.3% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 25.3%

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

6. Final simplification 25.3%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(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)))