b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 5.7s

Alternatives: 16

Speedup: 1.1×

• 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, 0.9× speedup

Math

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

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1e-8)
   (- (fma (/ m v) (fma -2.0 m 1.0) m) 1.0)
   (* (/ (* (- 1.0 m) m) v) (- 1.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1e-8) {
		tmp = fma((m / v), fma(-2.0, m, 1.0), m) - 1.0;
	} else {
		tmp = (((1.0 - m) * m) / v) * (1.0 - m);
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1e-8)
		tmp = Float64(fma(Float64(m / v), fma(-2.0, m, 1.0), m) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) * m) / v) * Float64(1.0 - m));
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1e-8], N[(N[(N[(m / v), $MachinePrecision] * N[(-2.0 * m + 1.0), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1e-8

   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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. distribute-lft-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1e-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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 2: 72.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

   5. Applied rewrites 98.4%

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

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

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 28.8%

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

   11. Taylor expanded in m around 0

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

   13. Applied rewrites 68.0%

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

4. Final simplification 75.6%

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

Alternative 3: 72.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

   5. Applied rewrites 98.4%

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

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

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 30.1

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

   7. Applied rewrites 30.1%

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

   8. Taylor expanded in m around inf

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

   10. Applied rewrites 29.0%

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

   11. Taylor expanded in m around 0

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

   13. Applied rewrites 68.0%

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

4. Final simplification 75.6%

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

Alternative 4: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 5: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. distribute-lft-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. 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. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.9%

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

Alternative 6: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1.6499999999999999

   1. Initial program 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. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 1.6499999999999999 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.5%

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

Alternative 7: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.65) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} 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 <= 1.65d0) then
        tmp = ((m / v) - 1.0d0) * (1.0d0 - m)
    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 <= 1.65) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((m / v) * m) * (m - 2.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1.6499999999999999

   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. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 1.6499999999999999 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

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

   5. Applied rewrites 99.8%

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

Alternative 8: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.43) (* (- (/ m v) 1.0) (- 1.0 m)) (/ (* (* m m) m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((m * m) * m) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 0.43d0) then
        tmp = ((m / v) - 1.0d0) * (1.0d0 - m)
    else
        tmp = ((m * m) * m) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.43) {
		tmp = ((m / v) - 1.0) * (1.0 - m);
	} else {
		tmp = ((m * m) * m) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.43:
		tmp = ((m / v) - 1.0) * (1.0 - m)
	else:
		tmp = ((m * m) * m) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.43)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * Float64(1.0 - m));
	else
		tmp = Float64(Float64(Float64(m * m) * m) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.43)
		tmp = ((m / v) - 1.0) * (1.0 - m);
	else
		tmp = ((m * m) * m) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.43], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(m * m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.429999999999999993

   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. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.429999999999999993 < 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. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in m around inf

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

   10. Applied rewrites 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.43:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 10: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.55:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.55) (- (+ (/ m v) m) 1.0) (/ (* (* m m) m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.55) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = ((m * m) * m) / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.55d0) then
        tmp = ((m / v) + m) - 1.0d0
    else
        tmp = ((m * m) * m) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.55) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = ((m * m) * m) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.55:
		tmp = ((m / v) + m) - 1.0
	else:
		tmp = ((m * m) * m) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.55)
		tmp = Float64(Float64(Float64(m / v) + m) - 1.0);
	else
		tmp = Float64(Float64(Float64(m * m) * m) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.55)
		tmp = ((m / v) + m) - 1.0;
	else
		tmp = ((m * m) * m) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.55], N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(m * m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5499999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2.5499999999999998 < 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. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in m around inf

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

   10. Applied rewrites 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.55:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.55:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.55) (- (+ (/ m v) m) 1.0) (* (* m m) (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.55) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = (m * m) * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.55d0) then
        tmp = ((m / v) + m) - 1.0d0
    else
        tmp = (m * m) * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.55) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = (m * m) * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.55:
		tmp = ((m / v) + m) - 1.0
	else:
		tmp = (m * m) * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.55)
		tmp = Float64(Float64(Float64(m / v) + m) - 1.0);
	else
		tmp = Float64(Float64(m * m) * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.55)
		tmp = ((m / v) + m) - 1.0;
	else
		tmp = (m * m) * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.55], N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(m * m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5499999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2.5499999999999998 < 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. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in m around inf

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

   10. Applied rewrites 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.55:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.55:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v} \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.55) (- (+ (/ m v) m) 1.0) (* (/ (* m m) v) m)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.55) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = ((m * m) / v) * m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.55d0) then
        tmp = ((m / v) + m) - 1.0d0
    else
        tmp = ((m * m) / v) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.55) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = ((m * m) / v) * m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.55:
		tmp = ((m / v) + m) - 1.0
	else:
		tmp = ((m * m) / v) * m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.55)
		tmp = Float64(Float64(Float64(m / v) + m) - 1.0);
	else
		tmp = Float64(Float64(Float64(m * m) / v) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.55)
		tmp = ((m / v) + m) - 1.0;
	else
		tmp = ((m * m) / v) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.55], N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision] * m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5499999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2.5499999999999998 < 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. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in m around inf

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

   10. Applied rewrites 99.0%

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

   12. Applied rewrites 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.55:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v} \cdot m\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(m - -1\right) \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (- (+ (/ m v) m) 1.0) (* (- m -1.0) (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = (m - -1.0) * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.4d0) then
        tmp = ((m / v) + m) - 1.0d0
    else
        tmp = (m - (-1.0d0)) * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = ((m / v) + m) - 1.0;
	} else {
		tmp = (m - -1.0) * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = ((m / v) + m) - 1.0
	else:
		tmp = (m - -1.0) * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(Float64(Float64(m / v) + m) - 1.0);
	else
		tmp = Float64(Float64(m - -1.0) * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = ((m / v) + m) - 1.0;
	else
		tmp = (m - -1.0) * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(N[(N[(m / v), $MachinePrecision] + m), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(m - -1.0), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2.39999999999999991 < 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 0.1%

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

       1. lift--.f64 N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. metadata-eval N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. unpow1 N/A

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

       9. metadata-eval N/A

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

       10. sqrt-pow1 N/A

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

       11. pow2 N/A

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

       12. sqr-neg-rev N/A

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

       13. mul-1-neg N/A

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

       14. mul-1-neg N/A

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

       15. pow2 N/A

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

       16. sqrt-pow1 N/A

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

       17. metadata-eval N/A

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

       18. metadata-eval N/A

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

       19. unpow-prod-down N/A

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

       20. *-commutative N/A

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

       21. associate-*l* N/A

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

       22. metadata-eval N/A

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

       23. *-commutative N/A

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

       24. *-lft-identity N/A

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

       25. unpow1 N/A

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

       26. lower--.f64 77.4

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

   12. Applied rewrites 77.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\left(\frac{m}{v} + m\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(m - -1\right) \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. lower--.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. lower-+.f64 N/A

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

   9. lower-/.f64 76.8

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

5. Applied rewrites 76.8%

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

Alternative 15: 27.2% 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. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 27.7

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

5. Applied rewrites 27.7%

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

Alternative 16: 24.8% accurate, 31.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

5. Applied rewrites 25.2%

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

Reproduce

herbie shell --seed 2024312 
(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)))