b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 9.1s

Alternatives: 15

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 100.0%

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

Alternative 2: 73.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if ((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= -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(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -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) * (((m * (1.0 - m)) / v) + -1.0)) <= -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[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $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 93.9%

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

   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

      \[\leadsto \color{blue}{m \cdot \left(1 + \frac{1}{v}\right) - 1} \]
   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-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

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

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

      10. /-lowering-/.f64 62.3

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

   5. Simplified 62.3%

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

   6. Taylor expanded in v around 0

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

      1. /-lowering-/.f64 61.9

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

   8. Simplified 61.9%

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

4. Final simplification 69.6%

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

Alternative 3: 28.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if ((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= -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(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -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) * (((m * (1.0 - m)) / v) + -1.0)) <= -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[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $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 93.9%

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

   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 4.3

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

   5. Simplified 4.3%

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

   6. Taylor expanded in m around inf

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

   8. Simplified 5.1%

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

4. Final simplification 26.6%

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

Alternative 4: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(m, -2, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 8.8e-8)
   (fma (fma m -2.0 1.0) (/ m v) -1.0)
   (* (- 1.0 m) (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.8e-8) {
		tmp = fma(fma(m, -2.0, 1.0), (m / v), -1.0);
	} else {
		tmp = (1.0 - m) * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8.8e-8)
		tmp = fma(fma(m, -2.0, 1.0), Float64(m / v), -1.0);
	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, 8.8e-8], N[(N[(m * -2.0 + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + -1.0), $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 < 8.7999999999999994e-8

   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 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in v around 0

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. div-inv N/A

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

      11. /-lowering-/.f64 99.9

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

   10. Applied egg-rr 99.9%

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

   if 8.7999999999999994e-8 < 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 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 99.9

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

   5. Simplified 99.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 99.9

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(m, -2, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.04 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(m, -2, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m, m \cdot \left(m + -2\right), m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.04e-7)
   (fma (fma m -2.0 1.0) (/ m v) -1.0)
   (/ (fma m (* m (+ m -2.0)) m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.04e-7) {
		tmp = fma(fma(m, -2.0, 1.0), (m / v), -1.0);
	} else {
		tmp = fma(m, (m * (m + -2.0)), m) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.04e-7)
		tmp = fma(fma(m, -2.0, 1.0), Float64(m / v), -1.0);
	else
		tmp = Float64(fma(m, Float64(m * Float64(m + -2.0)), m) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.04e-7], N[(N[(m * -2.0 + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(m * N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision] + m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.04e-7

   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 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in v around 0

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. div-inv N/A

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

      11. /-lowering-/.f64 99.9

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

   10. Applied egg-rr 99.9%

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

   if 1.04e-7 < 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 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in m around 0

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

   8. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.04 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(m, -2, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m, m \cdot \left(m + -2\right), m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{m \cdot \left(1 + \left(-2 \cdot \frac{m}{v} + \frac{1}{v}\right)\right) - 1} \]
   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 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in v around 0

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. div-inv N/A

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

      11. /-lowering-/.f64 99.2

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

   10. Applied egg-rr 99.2%

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

   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

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

   5. Simplified 99.7%

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

4. Final simplification 99.5%

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

Alternative 7: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.41:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(m, -2, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.41) (fma (fma m -2.0 1.0) (/ m v) -1.0) (* m (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.41) {
		tmp = fma(fma(m, -2.0, 1.0), (m / v), -1.0);
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.41)
		tmp = fma(fma(m, -2.0, 1.0), Float64(m / v), -1.0);
	else
		tmp = Float64(m * Float64(m * Float64(m / v)));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.41], N[(N[(m * -2.0 + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(m * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.409999999999999976

   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 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in v around 0

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. div-inv N/A

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

      11. /-lowering-/.f64 99.2

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

   10. Applied egg-rr 99.2%

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

   if 0.409999999999999976 < 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

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

   5. Simplified 99.7%

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

   6. Taylor expanded in m around inf

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

   8. Simplified 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.41:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(m, -2, 1\right), \frac{m}{v}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 98.4

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

   5. Simplified 98.4%

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

   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

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

   5. Simplified 99.7%

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

   6. Taylor expanded in m around inf

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

   8. Simplified 98.6%

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

4. Final simplification 98.5%

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

Alternative 9: 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 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

   \[\leadsto \color{blue}{\left(m \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right) - 1\right)} \cdot \left(1 - m\right) \]
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. Final simplification 99.9%

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

Alternative 10: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.38) {
		tmp = -1.0 + (m / v);
	} 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 <= 0.38d0) then
        tmp = (-1.0d0) + (m / v)
    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 <= 0.38) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * (m * (m / v));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.38

   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-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

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

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

      10. /-lowering-/.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in v around 0

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

      1. /-lowering-/.f64 98.4

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

   8. Simplified 98.4%

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

   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

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

   5. Simplified 99.7%

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

   6. Taylor expanded in m around inf

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

   8. Simplified 98.6%

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

4. Final simplification 98.5%

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

Alternative 11: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m, m, -1\right)}{m + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.35e+154) (+ -1.0 (+ m (/ m v))) (/ (fma m m -1.0) (+ m 1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.35e+154) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = fma(m, m, -1.0) / (m + 1.0);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.35e+154)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(fma(m, m, -1.0) / Float64(m + 1.0));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.35e+154], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * m + -1.0), $MachinePrecision] / N[(m + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.35000000000000003e154

   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-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

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

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

      10. /-lowering-/.f64 69.8

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

   5. Simplified 69.8%

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

   if 1.35000000000000003e154 < m

   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 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 6.9

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

   5. Simplified 6.9%

      \[\leadsto \color{blue}{-1 + m} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. flip-+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(m, m, -1\right)}{m + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.4% 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 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-rgt-in N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

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

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

   10. /-lowering-/.f64 71.4

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

5. Simplified 71.4%

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

Alternative 13: 75.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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-rgt-in N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

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

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

   10. /-lowering-/.f64 71.4

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

5. Simplified 71.4%

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

6. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 71.4

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

8. Simplified 71.4%

   \[\leadsto -1 + \color{blue}{\frac{m}{v}} \]
9. Add Preprocessing

Alternative 14: 27.6% 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 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 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 26.0

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

5. Simplified 26.0%

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

6. Final simplification 26.0%

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

Alternative 15: 25.1% 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 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 23.4%

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

Reproduce

herbie shell --seed 2024198 
(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)))