b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 8.8s

Alternatives: 14

Speedup: 1.0×

• 41.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ m + \mathsf{fma}\left(\mathsf{fma}\left(m, m + -2, 1\right), \frac{m}{v}, -1\right) \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (+ m (fma (fma m (+ m -2.0) 1.0) (/ m v) -1.0)))

C

double code(double m, double v) {
	return m + fma(fma(m, (m + -2.0), 1.0), (m / v), -1.0);
}

Julia

function code(m, v)
	return Float64(m + fma(fma(m, Float64(m + -2.0), 1.0), Float64(m / v), -1.0))
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. 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. associate-+r+ N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. associate--l+ N/A

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

   6. *-lft-identity N/A

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

   7. associate-*l/ N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.9%

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

6. Taylor expanded in m around 0

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. +-lowering-+.f64 N/A

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

   7. *-commutative N/A

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

8. Simplified 100.0%

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

Alternative 2: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -0.2:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- 1.0 m) (+ -1.0 (/ (* m (- 1.0 m)) v))) -0.2) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (((1.0 - m) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.2) {
		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 (((1.0d0 - m) * ((-1.0d0) + ((m * (1.0d0 - m)) / v))) <= (-0.2d0)) 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 (((1.0 - m) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.2) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if ((1.0 - m) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.2:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v))) <= -0.2)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (((1.0 - m) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.2)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.2], -1.0, N[(m / v), $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)) < -0.20000000000000001

   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. Simplified 93.6%

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

   if -0.20000000000000001 < (*.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. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate--l+ N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in v around 0

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

   8. Simplified 98.0%

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

   9. Taylor expanded in m around 0

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

       1. /-lowering-/.f64 68.3

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

   11. Simplified 68.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -0.2:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 27.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -0.2:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- 1.0 m) (+ -1.0 (/ (* m (- 1.0 m)) v))) -0.2) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (((1.0 - m) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.2) {
		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 (((1.0d0 - m) * ((-1.0d0) + ((m * (1.0d0 - m)) / v))) <= (-0.2d0)) 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 (((1.0 - m) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.2) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if ((1.0 - m) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.2:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(1.0 - m) * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v))) <= -0.2)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (((1.0 - m) * (-1.0 + ((m * (1.0 - m)) / v))) <= -0.2)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.2], -1.0, m]

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)) < -0.20000000000000001

   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. Simplified 93.6%

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

   if -0.20000000000000001 < (*.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 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. neg-sub0 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 3.8

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

   5. Simplified 3.8%

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

   6. Taylor expanded in m around inf

      \[\leadsto \color{blue}{m} \]
   7. Step-by-step derivation

   8. Simplified 5.0%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -0.2:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.4 \cdot 10^{-9}:\\ \;\;\;\;-1 + \mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 9.4e-9)
   (+ -1.0 (fma (/ m v) (fma m -2.0 1.0) m))
   (* (- 1.0 m) (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 9.4e-9) {
		tmp = -1.0 + fma((m / v), fma(m, -2.0, 1.0), m);
	} else {
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 9.4e-9)
		tmp = Float64(-1.0 + fma(Float64(m / v), fma(m, -2.0, 1.0), m));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 9.4e-9], N[(-1.0 + N[(N[(m / v), $MachinePrecision] * N[(m * -2.0 + 1.0), $MachinePrecision] + m), $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 < 9.3999999999999998e-9

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. distribute-rgt-out N/A

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

      14. *-rgt-identity N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. *-commutative N/A

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

      18. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   if 9.3999999999999998e-9 < 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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. div-sub N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in m around inf

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

      1. distribute-lft-out-- N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. rgt-mult-inverse N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-out-- N/A

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

      13. sub-neg N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. +-commutative N/A

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

      18. mul-1-neg N/A

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

      19. sub-neg N/A

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

      20. div-sub N/A

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

      21. /-lowering-/.f64 N/A

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

   8. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.4 \cdot 10^{-9}:\\ \;\;\;\;-1 + \mathsf{fma}\left(\frac{m}{v}, \mathsf{fma}\left(m, -2, 1\right), m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.05 \cdot 10^{-21}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(m, m + -2, 1\right) \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.05e-21)
   (+ -1.0 (+ m (/ m v)))
   (* (fma m (+ m -2.0) 1.0) (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.05e-21) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = fma(m, (m + -2.0), 1.0) * (m / v);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.05e-21)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(fma(m, Float64(m + -2.0), 1.0) * Float64(m / v));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.05e-21], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m + -2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.05000000000000006e-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. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.05000000000000006e-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. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate--l+ N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in v around 0

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

   8. Simplified 99.5%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(m + -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- 1.0 m) (+ (/ m v) -1.0)) (* (/ m v) (* m (+ m -2.0)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m / v) * (m * (m + -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.6d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = (m / v) * (m * (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = (m / v) * (m * (m + -2.0));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = (m / v) * (m * (m + -2.0))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m / v) * Float64(m * Float64(m + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = (m / v) * (m * (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   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

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

      1. /-lowering-/.f64 98.2

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

   5. Simplified 98.2%

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

   if 1.6000000000000001 < 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. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate--l+ N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in v around 0

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

   8. Simplified 99.8%

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

   9. Taylor expanded in m around inf

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

       1. unpow2 N/A

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

       2. sub-neg N/A

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

       3. +-commutative N/A

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

       4. neg-sub0 N/A

          \[\leadsto \left(\left(m \cdot m\right) \cdot \left(\color{blue}{\left(0 - 2 \cdot \frac{1}{m}\right)} + 1\right)\right) \cdot \frac{m}{v} \]

       5. associate-+l- N/A

          \[\leadsto \left(\left(m \cdot m\right) \cdot \color{blue}{\left(0 - \left(2 \cdot \frac{1}{m} - 1\right)\right)}\right) \cdot \frac{m}{v} \]

       6. neg-sub0 N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. *-lowering-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

       14. +-commutative N/A

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

       15. distribute-rgt-in N/A

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

       16. neg-mul-1 N/A

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

       17. mul-1-neg N/A

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

       18. remove-double-neg N/A

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

       19. associate-*l* N/A

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

       20. mul-1-neg N/A

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

       21. distribute-rgt-neg-out N/A

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

       22. lft-mult-inverse N/A

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

       23. metadata-eval N/A

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

   11. Simplified 98.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(m + -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(m + -2\right) \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6) (* (- 1.0 m) (+ (/ m v) -1.0)) (* m (* (+ m -2.0) (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m * ((m + -2.0) * (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 <= 1.6d0) then
        tmp = (1.0d0 - m) * ((m / v) + (-1.0d0))
    else
        tmp = m * ((m + (-2.0d0)) * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6) {
		tmp = (1.0 - m) * ((m / v) + -1.0);
	} else {
		tmp = m * ((m + -2.0) * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6:
		tmp = (1.0 - m) * ((m / v) + -1.0)
	else:
		tmp = m * ((m + -2.0) * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m * Float64(Float64(m + -2.0) * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = (1.0 - m) * ((m / v) + -1.0);
	else
		tmp = m * ((m + -2.0) * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m + -2.0), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   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

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

      1. /-lowering-/.f64 98.2

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

   5. Simplified 98.2%

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

   if 1.6000000000000001 < 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. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate--l+ N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. +-lowering-+.f64 N/A

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

      7. *-commutative N/A

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

   8. Simplified 99.9%

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

   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. Simplified 98.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(m + -2\right) \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      1. /-lowering-/.f64 98.2

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

   5. Simplified 98.2%

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

   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 inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

         \[\leadsto \frac{m \cdot \color{blue}{{m}^{2}}}{v} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{m \cdot {m}^{2}}}{v} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 96.9

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

   5. Simplified 96.9%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 96.9

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

   7. Applied egg-rr 96.9%

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

4. Final simplification 97.6%

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

Alternative 9: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. 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. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. div-sub N/A

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

   8. /-lowering-/.f64 N/A

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

   9. --lowering--.f64 99.8

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

5. Simplified 99.8%

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

   1. *-lowering-*.f64 N/A

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

   2. *-commutative N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. associate-/r/ N/A

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

   6. clear-num N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. --lowering--.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. --lowering--.f64 99.9

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 10: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. 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. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. div-sub N/A

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

   8. /-lowering-/.f64 N/A

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

   9. --lowering--.f64 99.8

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 11: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.38) (+ -1.0 (+ m (/ m v))) (* (/ m v) (* m m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 0.38d0) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m / v) * (m * m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m / v) * (m * m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.38:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m / v) * (m * m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.38)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m / v) * Float64(m * m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.38)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m / v) * (m * m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.38], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.38

   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. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 98.8

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

   5. Simplified 98.8%

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

   if 0.38 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

         \[\leadsto \frac{m \cdot \color{blue}{{m}^{2}}}{v} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{m \cdot {m}^{2}}}{v} \]

      5. unpow2 N/A

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

      6. *-lowering-*.f64 96.3

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

   5. Simplified 96.3%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 96.3

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

   7. Applied egg-rr 96.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(-1.0 + N[(m + N[(m / 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. Taylor expanded in m around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. +-lowering-+.f64 N/A

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

   10. /-lowering-/.f64 76.7

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

5. Simplified 76.7%

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

Alternative 13: 26.7% accurate, 7.8× 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. 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. neg-sub0 N/A

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

   3. associate--r- N/A

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

   4. metadata-eval N/A

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

   5. +-lowering-+.f64 23.8

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

5. Simplified 23.8%

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

6. Final simplification 23.8%

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

Alternative 14: 24.2% 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. Simplified 21.5%

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

Reproduce

herbie shell --seed 2024201 
(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)))