b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.8%

Time: 5.1s

Alternatives: 10

Speedup: N/A×

• 42.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.4 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m - 2, m, 1\right) \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.4e-21) (- (+ (/ m v) m) 1.0) (* (fma (- m 2.0) m 1.0) (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.4e-21) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = fma((m - 2.0), m, 1.0) * (m / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.4e-21)
		tmp = Float64(Float64(Float64(m / v) + m) - 1.0);
	else
		tmp = Float64(fma(Float64(m - 2.0), m, 1.0) * Float64(m / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.4e-21], N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(m - 2.0), $MachinePrecision] * m + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.4000000000000001e-21

   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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 4.4000000000000001e-21 < 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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(m, m, \mathsf{fma}\left(-2, m, 1\right)\right)}{v}, m, m - 1\right)} \]

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 99.9%

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

Alternative 2: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)) -0.5)
   -1.0
   (* 1.0 (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * (1.0 - m)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = 1.0 * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (((((m * (1.0d0 - m)) / v) - 1.0d0) * (1.0d0 - m)) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * (1.0 - m)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = 1.0 * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if ((((m * (1.0 - m)) / v) - 1.0) * (1.0 - m)) <= -0.5:
		tmp = -1.0
	else:
		tmp = 1.0 * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * Float64(1.0 - m)) <= -0.5)
		tmp = -1.0;
	else
		tmp = Float64(1.0 * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (((((m * (1.0 - m)) / v) - 1.0) * (1.0 - m)) <= -0.5)
		tmp = -1.0;
	else
		tmp = 1.0 * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(1.0 * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) m)) < -0.5

   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

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.2%

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

   if -0.5 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) (-.f64 #s(literal 1 binary64) 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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(m, m, \mathsf{fma}\left(-2, m, 1\right)\right)}{v}, m, m - 1\right)} \]

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in m around 0

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

   11. Applied rewrites 62.0%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.65:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(m - 2\right) \cdot m\right) \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.65) (* (- (/ m v) 1.0) (- 1.0 m)) (/ (* (* (- m 2.0) m) m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.65) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = (((m - 2.0) * m) * m) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.65d0) then
        tmp = ((m / v) - 1.0d0) * (1.0d0 - m)
    else
        tmp = (((m - 2.0d0) * m) * m) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.65) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = (((m - 2.0) * m) * m) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.65:
		tmp = ((m / v) - 1.0) * (1.0 - m)
	else:
		tmp = (((m - 2.0) * m) * m) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.65)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * Float64(1.0 - m));
	else
		tmp = Float64(Float64(Float64(Float64(m - 2.0) * m) * m) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.65)
		tmp = ((m / v) - 1.0) * (1.0 - m);
	else
		tmp = (((m - 2.0) * m) * m) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.65], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(m - 2.0), $MachinePrecision] * m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6499999999999999

   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

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

      1. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1.6499999999999999 < 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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(m, m, \mathsf{fma}\left(-2, m, 1\right)\right)}{v}, m, m - 1\right)} \]

   6. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]
   7. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

   8. Applied rewrites 98.6%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.65:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(m - 2\right) \cdot m\right) \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.65) (* (- (/ m v) 1.0) (- 1.0 m)) (* (* (- m 2.0) m) (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.65) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((m - 2.0) * m) * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.65d0) then
        tmp = ((m / v) - 1.0d0) * (1.0d0 - m)
    else
        tmp = ((m - 2.0d0) * m) * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.65) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((m - 2.0) * m) * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.65:
		tmp = ((m / v) - 1.0) * (1.0 - m)
	else:
		tmp = ((m - 2.0) * m) * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.65)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * Float64(1.0 - m));
	else
		tmp = Float64(Float64(Float64(m - 2.0) * m) * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.65)
		tmp = ((m / v) - 1.0) * (1.0 - m);
	else
		tmp = ((m - 2.0) * m) * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.65], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m - 2.0), $MachinePrecision] * m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6499999999999999

   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

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

      1. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1.6499999999999999 < 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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(m, m, \mathsf{fma}\left(-2, m, 1\right)\right)}{v}, m, m - 1\right)} \]

   6. Taylor expanded in m around inf

      \[\leadsto \mathsf{fma}\left(\frac{{m}^{2}}{v}, m, m - 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{v} - 2 \cdot \frac{1}{m \cdot v}\right)} \]

   11. Applied rewrites 98.6%

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

Alternative 5: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(m, m, \mathsf{fma}\left(-2, m, 1\right)\right)}{v}, m, m - 1\right)} \]

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 96.6%

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

Alternative 6: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. *-inverses N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-/.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 7: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6) (- (+ (/ m v) m) 1.0) (* (* m m) (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = (m * m) * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.6d0) then
        tmp = ((m / v) + m) - 1.0d0
    else
        tmp = (m * m) * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.6) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = (m * m) * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6:
		tmp = ((m / v) + m) - 1.0
	else:
		tmp = (m * m) * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6)
		tmp = Float64(Float64(Float64(m / v) + m) - 1.0);
	else
		tmp = Float64(Float64(m * m) * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6)
		tmp = ((m / v) + m) - 1.0;
	else
		tmp = (m * m) * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6], N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(m * m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000009

   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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 2.60000000000000009 < 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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(m, m, \mathsf{fma}\left(-2, m, 1\right)\right)}{v}, m, m - 1\right)} \]

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in m around inf

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

   11. Applied rewrites 96.6%

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

Alternative 8: 75.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in m around 0

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. lower--.f64 N/A

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 72.1

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

5. Applied rewrites 72.1%

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

Alternative 9: 26.3% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. *-lft-identity N/A

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

   3. metadata-eval N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. distribute-neg-in N/A

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

   6. metadata-eval N/A

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

   7. mul-1-neg N/A

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

   8. remove-double-neg N/A

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

   9. lower-+.f64 25.8

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

5. Applied rewrites 25.8%

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

Alternative 10: 23.8% accurate, 31.0× 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. Add Preprocessing

3. Taylor expanded in m around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 23.2%

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

Reproduce

herbie shell --seed 2024337 
(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)))