a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 5.4s

Alternatives: 11

Speedup: 1.0×

• 41.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+30}:\\ \;\;\;\;\frac{-m}{v} \cdot m\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* (- (/ (* m (- 1.0 m)) v) 1.0) m)))
   (if (<= t_0 -5e+30)
     (* (/ (- m) v) m)
     (if (<= t_0 -2e-308) (- m) (* (/ m v) m)))))

C

double code(double m, double v) {
	double t_0 = (((m * (1.0 - m)) / v) - 1.0) * m;
	double tmp;
	if (t_0 <= -5e+30) {
		tmp = (-m / v) * m;
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (((m * (1.0d0 - m)) / v) - 1.0d0) * m
    if (t_0 <= (-5d+30)) then
        tmp = (-m / v) * m
    else if (t_0 <= (-2d-308)) then
        tmp = -m
    else
        tmp = (m / v) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = (((m * (1.0 - m)) / v) - 1.0) * m;
	double tmp;
	if (t_0 <= -5e+30) {
		tmp = (-m / v) * m;
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = (((m * (1.0 - m)) / v) - 1.0) * m
	tmp = 0
	if t_0 <= -5e+30:
		tmp = (-m / v) * m
	elif t_0 <= -2e-308:
		tmp = -m
	else:
		tmp = (m / v) * m
	return tmp

Julia

function code(m, v)
	t_0 = Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
	tmp = 0.0
	if (t_0 <= -5e+30)
		tmp = Float64(Float64(Float64(-m) / v) * m);
	elseif (t_0 <= -2e-308)
		tmp = Float64(-m);
	else
		tmp = Float64(Float64(m / v) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = (((m * (1.0 - m)) / v) - 1.0) * m;
	tmp = 0.0;
	if (t_0 <= -5e+30)
		tmp = (-m / v) * m;
	elseif (t_0 <= -2e-308)
		tmp = -m;
	else
		tmp = (m / v) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+30], N[(N[((-m) / v), $MachinePrecision] * m), $MachinePrecision], If[LessEqual[t$95$0, -2e-308], (-m), N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -4.9999999999999998e30

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 0.1%

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

   10. Applied rewrites 73.8%

       \[\leadsto \frac{-m}{v} \cdot m \]

   if -4.9999999999999998e30 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.7%

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

   3. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 89.8%

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

Alternative 3: 72.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-\sqrt{m \cdot m}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* (- (/ (* m (- 1.0 m)) v) 1.0) m)))
   (if (<= t_0 (- INFINITY))
     (- (sqrt (* m m)))
     (if (<= t_0 -2e-308) (- m) (* (/ m v) m)))))

C

double code(double m, double v) {
	double t_0 = (((m * (1.0 - m)) / v) - 1.0) * m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -sqrt((m * m));
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	return tmp;
}

Java

public static double code(double m, double v) {
	double t_0 = (((m * (1.0 - m)) / v) - 1.0) * m;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -Math.sqrt((m * m));
	} else if (t_0 <= -2e-308) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = (((m * (1.0 - m)) / v) - 1.0) * m
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -math.sqrt((m * m))
	elif t_0 <= -2e-308:
		tmp = -m
	else:
		tmp = (m / v) * m
	return tmp

Julia

function code(m, v)
	t_0 = Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-sqrt(Float64(m * m)));
	elseif (t_0 <= -2e-308)
		tmp = Float64(-m);
	else
		tmp = Float64(Float64(m / v) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = (((m * (1.0 - m)) / v) - 1.0) * m;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -sqrt((m * m));
	elseif (t_0 <= -2e-308)
		tmp = -m;
	else
		tmp = (m / v) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-N[Sqrt[N[(m * m), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$0, -2e-308], (-m), N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.7

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

   5. Applied rewrites 5.7%

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

   7. Applied rewrites 60.6%

      \[\leadsto -\sqrt{m \cdot m} \]

   if -inf.0 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.7%

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

   3. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 89.8%

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

Alternative 4: 49.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* (- (/ (* m (- 1.0 m)) v) 1.0) m) -2e-308) (- m) (* (/ m v) m)))

C

double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -2e-308) {
		tmp = -m;
	} 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 (((((m * (1.0d0 - m)) / v) - 1.0d0) * m) <= (-2d-308)) then
        tmp = -m
    else
        tmp = (m / v) * m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -2e-308) {
		tmp = -m;
	} else {
		tmp = (m / v) * m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if ((((m * (1.0 - m)) / v) - 1.0) * m) <= -2e-308:
		tmp = -m
	else:
		tmp = (m / v) * m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m) <= -2e-308)
		tmp = Float64(-m);
	else
		tmp = Float64(Float64(m / v) * m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (((((m * (1.0 - m)) / v) - 1.0) * m) <= -2e-308)
		tmp = -m;
	else
		tmp = (m / v) * m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], -2e-308], (-m), 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)) m) < -1.9999999999999998e-308

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 35.8

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

   5. Applied rewrites 35.8%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.7%

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

   3. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 89.8%

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

Alternative 5: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{m - v}{v} \cdot m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - m\right) \cdot m\right) \cdot m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7.7e-16) (* (/ (- m v) v) m) (/ (* (* (- 1.0 m) m) m) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.7e-16) {
		tmp = ((m - v) / v) * m;
	} else {
		tmp = (((1.0 - 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 <= 7.7d-16) then
        tmp = ((m - v) / v) * m
    else
        tmp = (((1.0d0 - m) * m) * m) / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7.7e-16) {
		tmp = ((m - v) / v) * m;
	} else {
		tmp = (((1.0 - m) * m) * m) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.7e-16:
		tmp = ((m - v) / v) * m
	else:
		tmp = (((1.0 - m) * m) * m) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.7e-16)
		tmp = Float64(Float64(Float64(m - v) / v) * m);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - m) * m) * m) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7.7e-16)
		tmp = ((m - v) / v) * m;
	else
		tmp = (((1.0 - m) * m) * m) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.7e-16], N[(N[(N[(m - v), $MachinePrecision] / v), $MachinePrecision] * m), $MachinePrecision], N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.69999999999999989e-16

   1. Initial program 99.8%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in m around 0

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

      1. distribute-lft-in N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. *-inverses N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. +-commutative N/A

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

      13. div-add N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 99.8%

       \[\leadsto \frac{m - v}{v} \cdot m \]

   if 7.69999999999999989e-16 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.9%

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

Alternative 6: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{m - v}{v} \cdot m\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7.7e-16) (* (/ (- m v) v) m) (* (/ (- 1.0 m) v) (* m m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.7e-16) {
		tmp = ((m - v) / v) * m;
	} else {
		tmp = ((1.0 - m) / v) * (m * m);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 7.7d-16) then
        tmp = ((m - v) / v) * m
    else
        tmp = ((1.0d0 - m) / v) * (m * m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7.7e-16) {
		tmp = ((m - v) / v) * m;
	} else {
		tmp = ((1.0 - m) / v) * (m * m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.7e-16:
		tmp = ((m - v) / v) * m
	else:
		tmp = ((1.0 - m) / v) * (m * m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.7e-16)
		tmp = Float64(Float64(Float64(m - v) / v) * m);
	else
		tmp = Float64(Float64(Float64(1.0 - m) / v) * Float64(m * m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7.7e-16)
		tmp = ((m - v) / v) * m;
	else
		tmp = ((1.0 - m) / v) * (m * m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.7e-16], N[(N[(N[(m - v), $MachinePrecision] / v), $MachinePrecision] * m), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.69999999999999989e-16

   1. Initial program 99.8%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in m around 0

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

      1. distribute-lft-in N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. *-inverses N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. +-commutative N/A

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

      13. div-add N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 99.8%

       \[\leadsto \frac{m - v}{v} \cdot m \]

   if 7.69999999999999989e-16 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 7: 97.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in m around 0

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

      1. distribute-lft-in N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. *-inverses N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. +-commutative N/A

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

      13. div-add N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 97.2%

       \[\leadsto \frac{m - v}{v} \cdot m \]

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 8: 87.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in m around 0

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

      1. distribute-lft-in N/A

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

      2. associate--l+ N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. *-inverses N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. +-commutative N/A

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

      13. div-add N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in m around 0

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

   10. Applied rewrites 97.2%

       \[\leadsto \frac{m - v}{v} \cdot m \]

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in m around 0

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

   8. Applied rewrites 0.1%

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

   10. Applied rewrites 73.8%

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

Alternative 9: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. +-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. *-inverses N/A

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

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

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

   11. *-inverses N/A

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

   12. metadata-eval N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower--.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 10: 27.0% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := (-m)

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 25.9

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

5. Applied rewrites 25.9%

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

Alternative 11: 3.1% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ m \end{array} \]

FPCore

(FPCore (m v) :precision binary64 m)

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return m

Julia

function code(m, v)
	return m
end

MATLAB

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

Wolfram

code[m_, v_] := m

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 25.9

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

5. Applied rewrites 25.9%

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

7. Applied rewrites 33.3%

   \[\leadsto -\sqrt{m \cdot m} \]
8. Step-by-step derivation

9. Applied rewrites 3.2%

   \[\leadsto \color{blue}{m} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(FPCore (m v)
  :name "a 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) m))