b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.8%

Time: 7.4s

Alternatives: 16

Speedup: 1.0×

• 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: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-14}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m + -1\right)}{v} \cdot \left(m + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.6e-14)
   (+ -1.0 (+ m (/ m v)))
   (* (/ (* m (+ m -1.0)) v) (+ m -1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.6e-14) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = ((m * (m + -1.0)) / v) * (m + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 3.6d-14) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = ((m * (m + (-1.0d0))) / v) * (m + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.6e-14) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = ((m * (m + -1.0)) / v) * (m + -1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.6e-14:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = ((m * (m + -1.0)) / v) * (m + -1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.6e-14)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(Float64(m * Float64(m + -1.0)) / v) * Float64(m + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.6e-14)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = ((m * (m + -1.0)) / v) * (m + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.6e-14], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m * N[(m + -1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] * N[(m + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.5999999999999998e-14

   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. Taylor expanded in m around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. distribute-lft-in 99.8%

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

      3. div-inv 100.0%

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

      4. *-rgt-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 3.5999999999999998e-14 < 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. associate-/l* 99.9%

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

      3. fmm-def 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in v around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-14}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m + -1\right)}{v} \cdot \left(m + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-14}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.5e-14)
   (+ -1.0 (+ m (/ m v)))
   (* (- 1.0 m) (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.5e-14) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (1.0 - m) * (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 <= 6.5d-14) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (1.0d0 - m) * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 6.5e-14) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.5e-14:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (1.0 - m) * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.5e-14)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.5e-14)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.5e-14], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.5000000000000001e-14

   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. Taylor expanded in m around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. distribute-lft-in 99.8%

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

      3. div-inv 100.0%

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

      4. *-rgt-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 6.5000000000000001e-14 < 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. associate-/l* 99.9%

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

      3. fmm-def 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in v around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-14}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 96.9%

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

   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. associate-/l* 99.9%

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

      3. fmm-def 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in v around 0 99.9%

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

   7. Taylor expanded in m around inf 97.7%

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

      1. neg-mul-1 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 97.3%

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

Alternative 4: 87.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 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 96.9%

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

   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. associate-/l* 99.9%

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

      3. fmm-def 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. neg-mul-1 0.1%

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

      2. unsub-neg 0.1%

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

   8. Simplified 0.1%

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

   9. Taylor expanded in m around 0 0.1%

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

       1. neg-mul-1 0.1%

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

       2. sub-neg 0.1%

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

       3. rem-square-sqrt 0.0%

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

       4. fabs-sqr 0.0%

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

       5. rem-square-sqrt 76.5%

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

       6. fabs-neg 76.5%

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

       7. rem-square-sqrt 76.5%

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

       8. fabs-sqr 76.5%

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

       9. rem-square-sqrt 76.5%

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

       10. neg-sub0 76.5%

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

       11. associate--r- 76.5%

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

       12. metadata-eval 76.5%

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

   11. Simplified 76.5%

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

   12. Taylor expanded in v around 0 76.5%

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

       1. sub-neg 76.5%

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

       2. metadata-eval 76.5%

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

       3. associate-/l* 76.5%

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

   14. Simplified 76.5%

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

4. Final simplification 86.4%

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

Alternative 5: 87.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 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. associate-/l* 99.7%

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

      3. fmm-def 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in m around 0 96.8%

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

      1. neg-mul-1 96.8%

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

      2. unsub-neg 96.8%

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

   8. Simplified 96.8%

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

   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. associate-/l* 99.9%

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

      3. fmm-def 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. neg-mul-1 0.1%

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

      2. unsub-neg 0.1%

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

   8. Simplified 0.1%

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

   9. Taylor expanded in m around 0 0.1%

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

       1. neg-mul-1 0.1%

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

       2. sub-neg 0.1%

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

       3. rem-square-sqrt 0.0%

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

       4. fabs-sqr 0.0%

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

       5. rem-square-sqrt 76.5%

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

       6. fabs-neg 76.5%

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

       7. rem-square-sqrt 76.5%

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

       8. fabs-sqr 76.5%

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

       9. rem-square-sqrt 76.5%

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

       10. neg-sub0 76.5%

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

       11. associate--r- 76.5%

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

       12. metadata-eval 76.5%

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

   11. Simplified 76.5%

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

   12. Taylor expanded in v around 0 76.5%

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

       1. sub-neg 76.5%

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

       2. metadata-eval 76.5%

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

       3. associate-/l* 76.5%

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

   14. Simplified 76.5%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 7: 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 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%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 87.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.7%

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

      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. Taylor expanded in m around 0 96.6%

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

      1. +-commutative 96.6%

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

      2. distribute-lft-in 96.6%

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

      3. div-inv 96.8%

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

      4. *-rgt-identity 96.8%

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

   7. Applied egg-rr 96.8%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fmm-def 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. neg-mul-1 0.1%

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

      2. unsub-neg 0.1%

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

   8. Simplified 0.1%

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

   9. Taylor expanded in m around 0 0.1%

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

       1. neg-mul-1 0.1%

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

       2. sub-neg 0.1%

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

       3. rem-square-sqrt 0.0%

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

       4. fabs-sqr 0.0%

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

       5. rem-square-sqrt 76.5%

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

       6. fabs-neg 76.5%

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

       7. rem-square-sqrt 76.5%

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

       8. fabs-sqr 76.5%

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

       9. rem-square-sqrt 76.5%

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

       10. neg-sub0 76.5%

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

       11. associate--r- 76.5%

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

       12. metadata-eval 76.5%

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

   11. Simplified 76.5%

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

   12. Taylor expanded in v around 0 76.5%

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

       1. sub-neg 76.5%

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

       2. metadata-eval 76.5%

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

       3. associate-/l* 76.5%

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

   14. Simplified 76.5%

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

4. Final simplification 86.3%

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

Alternative 10: 87.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.7%

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

      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. Taylor expanded in m around 0 96.6%

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

   6. Taylor expanded in v around 0 96.7%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fmm-def 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. neg-mul-1 0.1%

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

      2. unsub-neg 0.1%

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

   8. Simplified 0.1%

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

   9. Taylor expanded in m around 0 0.1%

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

       1. neg-mul-1 0.1%

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

       2. sub-neg 0.1%

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

       3. rem-square-sqrt 0.0%

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

       4. fabs-sqr 0.0%

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

       5. rem-square-sqrt 76.5%

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

       6. fabs-neg 76.5%

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

       7. rem-square-sqrt 76.5%

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

       8. fabs-sqr 76.5%

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

       9. rem-square-sqrt 76.5%

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

       10. neg-sub0 76.5%

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

       11. associate--r- 76.5%

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

       12. metadata-eval 76.5%

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

   11. Simplified 76.5%

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

   12. Taylor expanded in v around 0 76.5%

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

       1. sub-neg 76.5%

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

       2. metadata-eval 76.5%

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

       3. associate-/l* 76.5%

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

   14. Simplified 76.5%

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

4. Final simplification 86.3%

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

Alternative 11: 61.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{-193}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 3.5e-193) -1.0 (+ m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.5e-193) {
		tmp = -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 <= 3.5d-193) then
        tmp = -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 <= 3.5e-193) {
		tmp = -1.0;
	} else {
		tmp = m + (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.5e-193:
		tmp = -1.0
	else:
		tmp = m + (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.5e-193)
		tmp = -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 <= 3.5e-193)
		tmp = -1.0;
	else
		tmp = m + (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.5e-193], -1.0, N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.50000000000000005e-193

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 80.8%

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

   if 3.50000000000000005e-193 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 69.9%

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

   6. Taylor expanded in m around inf 61.1%

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

      1. +-commutative 69.9%

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

      2. distribute-lft-in 69.9%

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

      3. div-inv 70.0%

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

      4. *-rgt-identity 70.0%

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

   8. Applied egg-rr 61.2%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{-193}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.5 \cdot 10^{-193}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.5e-193) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.5e-193) {
		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 <= 2.5d-193) 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 <= 2.5e-193) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.5e-193:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.5e-193)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.5e-193)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.5e-193], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5000000000000002e-193

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 80.8%

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

   if 2.5000000000000002e-193 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 69.9%

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

   6. Taylor expanded in m around inf 61.1%

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

   7. Taylor expanded in v around 0 61.2%

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

Alternative 13: 27.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-14}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2e-14) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2e-14) {
		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 <= 2d-14) 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 <= 2e-14) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2e-14:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2e-14)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2e-14)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2e-14], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 2e-14

   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. Taylor expanded in m around 0 39.3%

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

   if 2e-14 < 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 0 53.2%

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

   6. Taylor expanded in v around inf 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 14: 75.7% 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 74.1%

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

6. Taylor expanded in v around 0 74.2%

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

7. Final simplification 74.2%

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

Alternative 15: 27.2% 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 m around 0 74.1%

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

6. Taylor expanded in v around inf 20.6%

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

7. Final simplification 20.6%

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

Alternative 16: 24.8% 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 18.1%

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

Reproduce

herbie shell --seed 2024175 
(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)))