b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.4s

Alternatives: 11

Speedup: N/A×

• 41.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% 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. metadata-eval 99.8%

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

   2. sub-neg 99.8%

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

   3. clear-num 99.8%

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

   4. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

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 / (v / m)));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) * (-1.0 - (m / (v / m)));
	}
	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 / (v / m)))
	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(v / m))));
	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 / (v / m)));
	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[(v / m), $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 98.6%

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

   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. Step-by-step derivation

      1. metadata-eval 99.9%

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

      2. sub-neg 99.9%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. neg-mul-1 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 99.1%

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

Alternative 3: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.44:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m}{\frac{v}{m}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.44) (* (- 1.0 m) (+ -1.0 (/ m v))) (* m (+ 1.0 (/ m (/ v m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.44) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * (1.0 + (m / (v / m)));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 0.44d0) then
        tmp = (1.0d0 - m) * ((-1.0d0) + (m / v))
    else
        tmp = m * (1.0d0 + (m / (v / m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.44) {
		tmp = (1.0 - m) * (-1.0 + (m / v));
	} else {
		tmp = m * (1.0 + (m / (v / m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.44:
		tmp = (1.0 - m) * (-1.0 + (m / v))
	else:
		tmp = m * (1.0 + (m / (v / m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.44)
		tmp = Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(1.0 + Float64(m / Float64(v / m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.44)
		tmp = (1.0 - m) * (-1.0 + (m / v));
	else
		tmp = m * (1.0 + (m / (v / m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.44], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(1.0 + N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.440000000000000002

   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 99.3%

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

   if 0.440000000000000002 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. metadata-eval 99.9%

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

      2. sub-neg 99.9%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. neg-mul-1 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in m around inf 98.9%

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

       1. neg-mul-1 98.9%

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

   12. Simplified 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.44:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + \frac{m}{\frac{v}{m}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

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(m * Float64(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[(m * N[(N[(1.0 - m), $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. Add Preprocessing

Alternative 5: 87.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.28) {
		tmp = -1.0 + (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 <= 0.28d0) then
        tmp = (-1.0d0) + (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 <= 0.28) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m * m) / v;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 0.28], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m * m), $MachinePrecision] / v), $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* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 99.1%

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

   6. Taylor expanded in m around 0 99.2%

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

   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 0.2%

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

      1. *-commutative 0.2%

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

      2. sub-neg 0.2%

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

      3. metadata-eval 0.2%

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

      4. distribute-lft-in 0.2%

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

      5. *-commutative 0.2%

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

      6. associate-*l/ 0.2%

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

      7. *-commutative 0.2%

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

      8. sub-neg 0.2%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 78.2%

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

      11. sqr-neg 78.2%

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

      12. sqrt-unprod 78.2%

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

      13. add-sqr-sqrt 78.2%

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

      14. +-commutative 78.2%

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

      15. distribute-lft1-in 78.2%

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

      16. fma-define 78.2%

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

      17. sub-neg 78.2%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 28.2%

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

      20. sqr-neg 28.2%

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

      21. sqrt-unprod 78.2%

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

      22. add-sqr-sqrt 78.2%

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

      23. +-commutative 78.2%

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

   5. Applied egg-rr 78.2%

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

      1. *-lft-identity 78.2%

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

      2. associate-*l/ 78.2%

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

      3. associate-*l/ 78.2%

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

      4. *-lft-identity 78.2%

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

      5. fma-undefine 78.2%

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

      6. distribute-lft1-in 78.2%

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

      7. associate-*r/ 78.2%

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

      8. distribute-lft-in 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in v around 0 78.2%

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

   9. Taylor expanded in m around inf 78.3%

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

4. Final simplification 88.6%

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

Alternative 6: 87.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0 49.0%

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

   1. *-commutative 49.0%

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

   2. sub-neg 49.0%

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

   3. metadata-eval 49.0%

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

   4. distribute-lft-in 49.0%

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

   5. *-commutative 49.0%

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

   6. associate-*l/ 49.0%

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

   7. *-commutative 49.0%

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

   8. sub-neg 49.0%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 88.6%

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

   11. sqr-neg 88.6%

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

   12. sqrt-unprod 88.6%

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

   13. add-sqr-sqrt 88.6%

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

   14. +-commutative 88.6%

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

   15. distribute-lft1-in 88.6%

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

   16. fma-define 88.6%

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

   17. sub-neg 88.6%

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

   18. add-sqr-sqrt 0.0%

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

   19. sqrt-unprod 63.2%

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

   20. sqr-neg 63.2%

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

   21. sqrt-unprod 88.6%

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

   22. add-sqr-sqrt 88.6%

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

   23. +-commutative 88.6%

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

5. Applied egg-rr 88.6%

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

   1. *-lft-identity 88.6%

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

   2. associate-*l/ 88.5%

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

   3. associate-*l/ 88.6%

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

   4. *-lft-identity 88.6%

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

   5. fma-undefine 88.6%

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

   6. distribute-lft1-in 88.6%

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

   7. associate-*r/ 88.6%

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

   8. distribute-lft-in 88.6%

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

7. Simplified 88.6%

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

8. Final simplification 88.6%

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

Alternative 7: 61.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.2 \cdot 10^{-103}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 8.2e-103) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.2e-103) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 8.2d-103) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 8.2e-103) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8.2e-103:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8.2e-103)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 8.2e-103)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 8.2e-103], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.19999999999999992e-103

   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%

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

      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 0 66.6%

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

   if 8.19999999999999992e-103 < 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 26.2%

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

      1. *-commutative 26.2%

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

      2. sub-neg 26.2%

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

      3. metadata-eval 26.2%

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

      4. distribute-lft-in 26.2%

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

      5. *-commutative 26.2%

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

      6. associate-*l/ 26.2%

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

      7. *-commutative 26.2%

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

      8. sub-neg 26.2%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 83.4%

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

      11. sqr-neg 83.4%

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

      12. sqrt-unprod 83.4%

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

      13. add-sqr-sqrt 83.4%

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

      14. +-commutative 83.4%

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

      15. distribute-lft1-in 83.4%

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

      16. fma-define 83.4%

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

      17. sub-neg 83.4%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 46.7%

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

      20. sqr-neg 46.7%

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

      21. sqrt-unprod 83.4%

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

      22. add-sqr-sqrt 83.4%

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

      23. +-commutative 83.4%

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

   5. Applied egg-rr 83.4%

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

      1. *-lft-identity 83.4%

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

      2. associate-*l/ 83.4%

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

      3. associate-*l/ 83.4%

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

      4. *-lft-identity 83.4%

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

      5. fma-undefine 83.4%

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

      6. distribute-lft1-in 83.4%

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

      7. associate-*r/ 83.4%

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

      8. distribute-lft-in 83.4%

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

   7. Simplified 83.4%

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

   8. Taylor expanded in v around 0 79.9%

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

   9. Taylor expanded in m around 0 65.0%

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

Alternative 8: 26.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.7 \cdot 10^{-25}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 5.7e-25) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.7e-25) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 5.7d-25) then
        tmp = -1.0d0
    else
        tmp = m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5.7e-25) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.7e-25:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.7e-25)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.7e-25)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.7e-25], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 5.7000000000000004e-25

   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 50.1%

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

   if 5.7000000000000004e-25 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 91.7%

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

      1. neg-mul-1 92.4%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in m around 0 5.3%

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

   7. Taylor expanded in m around inf 5.6%

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

Alternative 9: 76.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 48.9%

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

6. Taylor expanded in m around 0 78.3%

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

7. Final simplification 78.3%

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

Alternative 10: 26.5% 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 25.6%

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

   1. neg-mul-1 25.6%

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

   2. neg-sub0 25.6%

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

   3. associate--r- 25.6%

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

   4. metadata-eval 25.6%

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

7. Simplified 25.6%

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

8. Final simplification 25.6%

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

Alternative 11: 24.1% 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 23.1%

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

Reproduce

herbie shell --seed 2024155 
(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)))