b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.9s

Alternatives: 15

Speedup: 1.0×

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

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

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)) * ((2.0d0 - m) + (-1.0d0))
end function

Java

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

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) + -1.0) * ((2.0 - m) + -1.0)

Julia

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

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) + -1.0) * ((2.0 - m) + -1.0);
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. expm1-log1p-u 53.9%

      \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]

4. Applied egg-rr 53.9%

   \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - m\right)\right)} \]
5. Step-by-step derivation

   1. expm1-undefine 53.9%

      \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} - 1\right)} \]

   2. sub-neg 53.9%

      \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - m\right)} + \left(-1\right)\right)} \]

   3. log1p-undefine 53.9%

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

   4. rem-exp-log 99.9%

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

   5. associate-+r- 99.9%

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

   6. metadata-eval 99.9%

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

   7. metadata-eval 99.9%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 97.3%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 97.7%

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

      1. neg-mul-1 97.7%

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

   5. Simplified 97.7%

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

4. Final simplification 97.5%

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

Alternative 3: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 97.3%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around inf 97.7%

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

      1. neg-mul-1 97.7%

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

   7. Simplified 97.7%

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

4. Final simplification 97.5%

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

Alternative 4: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{-94}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.4e-94) -1.0 (if (<= m 0.27) (/ m v) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.4e-94) {
		tmp = -1.0;
	} else if (m <= 0.27) {
		tmp = m / v;
	} else {
		tmp = 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 <= 1.4d-94) then
        tmp = -1.0d0
    else if (m <= 0.27d0) then
        tmp = m / v
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.4e-94) {
		tmp = -1.0;
	} else if (m <= 0.27) {
		tmp = m / v;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.4e-94:
		tmp = -1.0
	elif m <= 0.27:
		tmp = m / v
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.4e-94)
		tmp = -1.0;
	elseif (m <= 0.27)
		tmp = Float64(m / v);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.4e-94)
		tmp = -1.0;
	elseif (m <= 0.27)
		tmp = m / v;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.4e-94], -1.0, If[LessEqual[m, 0.27], N[(m / v), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.3999999999999999e-94

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 67.6%

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

   if 1.3999999999999999e-94 < m < 0.27000000000000002

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 93.0%

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

   4. Taylor expanded in v around 0 83.5%

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

      1. associate-/l* 83.1%

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

   6. Simplified 83.1%

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

   7. Taylor expanded in m around 0 83.1%

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

   if 0.27000000000000002 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. sub-neg 0.1%

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

      2. distribute-lft-in 0.1%

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

      3. *-commutative 0.1%

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

      4. *-un-lft-identity 0.1%

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

      5. sub-neg 0.1%

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

      6. metadata-eval 0.1%

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

      7. +-commutative 0.1%

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

      8. sub-neg 0.1%

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

      9. metadata-eval 0.1%

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

      10. +-commutative 0.1%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 79.2%

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

      13. sqr-neg 79.2%

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

      14. sqrt-unprod 79.2%

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

      15. add-sqr-sqrt 79.2%

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

   5. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. distribute-rgt1-in 79.2%

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

      3. +-commutative 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in v around 0 79.2%

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

      1. +-commutative 79.2%

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

      2. associate-*l/ 79.2%

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

   10. Simplified 79.2%

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

   11. Taylor expanded in m around inf 79.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.4 \cdot 10^{-94}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 0.43], N[(N[(1.0 - m), $MachinePrecision] * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $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 97.3%

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

   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. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around inf 97.7%

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

      1. neg-mul-1 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in m around inf 97.6%

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

      1. neg-mul-1 97.7%

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

   10. Simplified 97.6%

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

4. Final simplification 97.4%

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

Alternative 6: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.35:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{1}{v}\right) \cdot \left(m + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.35) (+ -1.0 (/ m v)) (* (* m (/ 1.0 v)) (+ m 1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.35) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m * (1.0 / v)) * (m + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.35d0) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = (m * (1.0d0 / v)) * (m + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.35) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m * (1.0 / v)) * (m + 1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.35:
		tmp = -1.0 + (m / v)
	else:
		tmp = (m * (1.0 / v)) * (m + 1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.35)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(Float64(m * Float64(1.0 / v)) * Float64(m + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.35)
		tmp = -1.0 + (m / v);
	else
		tmp = (m * (1.0 / v)) * (m + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.35], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(1.0 / v), $MachinePrecision]), $MachinePrecision] * N[(m + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.35000000000000009

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 97.0%

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

   6. Taylor expanded in m around 0 97.0%

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

   if 2.35000000000000009 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. sub-neg 0.1%

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

      2. distribute-lft-in 0.1%

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

      3. *-commutative 0.1%

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

      4. *-un-lft-identity 0.1%

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

      5. sub-neg 0.1%

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

      6. metadata-eval 0.1%

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

      7. +-commutative 0.1%

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

      8. sub-neg 0.1%

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

      9. metadata-eval 0.1%

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

      10. +-commutative 0.1%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 79.2%

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

      13. sqr-neg 79.2%

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

      14. sqrt-unprod 79.2%

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

      15. add-sqr-sqrt 79.2%

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

   5. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. distribute-rgt1-in 79.2%

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

      3. +-commutative 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in v around 0 79.2%

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

      1. +-commutative 79.2%

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

      2. associate-*l/ 79.2%

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

   10. Simplified 79.2%

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

       1. clear-num 79.2%

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

       2. associate-/r/ 79.2%

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

   12. Applied egg-rr 79.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.35:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{1}{v}\right) \cdot \left(m + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. distribute-rgt-in 99.9%

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

   3. *-un-lft-identity 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 8: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 10: 87.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.35:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.35) (+ -1.0 (/ m v)) (* (/ m v) (+ m 1.0))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.35) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m / v) * (m + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.35d0) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = (m / v) * (m + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.35) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m / v) * (m + 1.0);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.35:
		tmp = -1.0 + (m / v)
	else:
		tmp = (m / v) * (m + 1.0)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.35)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(Float64(m / v) * Float64(m + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.35)
		tmp = -1.0 + (m / v);
	else
		tmp = (m / v) * (m + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.35], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(m + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.35000000000000009

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 97.0%

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

   6. Taylor expanded in m around 0 97.0%

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

   if 2.35000000000000009 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. sub-neg 0.1%

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

      2. distribute-lft-in 0.1%

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

      3. *-commutative 0.1%

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

      4. *-un-lft-identity 0.1%

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

      5. sub-neg 0.1%

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

      6. metadata-eval 0.1%

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

      7. +-commutative 0.1%

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

      8. sub-neg 0.1%

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

      9. metadata-eval 0.1%

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

      10. +-commutative 0.1%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 79.2%

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

      13. sqr-neg 79.2%

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

      14. sqrt-unprod 79.2%

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

      15. add-sqr-sqrt 79.2%

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

   5. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. distribute-rgt1-in 79.2%

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

      3. +-commutative 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in v around 0 79.2%

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

      1. +-commutative 79.2%

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

      2. associate-*l/ 79.2%

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

   10. Simplified 79.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.35:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.27:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 0.27) (+ -1.0 (/ m v)) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 0.27) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = 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 <= 0.27d0) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 0.27) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 0.27:
		tmp = -1.0 + (m / v)
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 0.27)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 0.27)
		tmp = -1.0 + (m / v);
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 0.27], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.27000000000000002

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 97.0%

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

   6. Taylor expanded in m around 0 97.0%

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

   if 0.27000000000000002 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. sub-neg 0.1%

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

      2. distribute-lft-in 0.1%

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

      3. *-commutative 0.1%

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

      4. *-un-lft-identity 0.1%

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

      5. sub-neg 0.1%

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

      6. metadata-eval 0.1%

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

      7. +-commutative 0.1%

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

      8. sub-neg 0.1%

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

      9. metadata-eval 0.1%

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

      10. +-commutative 0.1%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 79.2%

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

      13. sqr-neg 79.2%

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

      14. sqrt-unprod 79.2%

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

      15. add-sqr-sqrt 79.2%

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

   5. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. distribute-rgt1-in 79.2%

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

      3. +-commutative 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in v around 0 79.2%

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

      1. +-commutative 79.2%

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

      2. associate-*l/ 79.2%

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

   10. Simplified 79.2%

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

   11. Taylor expanded in m around inf 79.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.27:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in m around 0 52.5%

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

   1. sub-neg 52.5%

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

   2. distribute-lft-in 52.5%

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

   3. *-commutative 52.5%

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

   4. *-un-lft-identity 52.5%

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

   5. sub-neg 52.5%

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

   6. metadata-eval 52.5%

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

   7. +-commutative 52.5%

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

   8. sub-neg 52.5%

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

   9. metadata-eval 52.5%

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

   10. +-commutative 52.5%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 88.8%

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

   13. sqr-neg 88.8%

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

   14. sqrt-unprod 88.8%

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

   15. add-sqr-sqrt 88.8%

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

5. Applied egg-rr 88.8%

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

   1. *-commutative 88.8%

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

   2. distribute-rgt1-in 88.8%

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

   3. +-commutative 88.8%

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

7. Simplified 88.8%

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

8. Final simplification 88.8%

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

Alternative 13: 60.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-94}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.5e-94) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.5e-94) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.5d-94) then
        tmp = -1.0d0
    else
        tmp = m / v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.5e-94) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.5e-94:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.5e-94)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.5e-94)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.5e-94], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.5000000000000001e-94

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 67.6%

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

   if 1.5000000000000001e-94 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 28.9%

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

   4. Taylor expanded in v around 0 25.9%

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

      1. associate-/l* 25.8%

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

   6. Simplified 25.8%

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

   7. Taylor expanded in m around 0 64.5%

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

Alternative 14: 26.6% accurate, 4.3× 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. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 27.0%

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

   1. neg-mul-1 27.0%

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

   2. neg-sub0 27.0%

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

   3. associate--r- 27.0%

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

   4. metadata-eval 27.0%

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

7. Simplified 27.0%

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

8. Final simplification 27.0%

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

Alternative 15: 24.1% accurate, 13.0× 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. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 24.7%

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

Reproduce

herbie shell --seed 2024180 
(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)))