b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 13

Speedup: 1.0×

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

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $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. 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* 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.3 \cdot 10^{-32}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.3e-32) (+ -1.0 (/ m v)) (* (- 1.0 m) (/ m (/ v (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.3e-32) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (1.0 - m) * (m / (v / (1.0 - 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 <= 4.3d-32) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = (1.0d0 - m) * (m / (v / (1.0d0 - m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.3e-32) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (1.0 - m) * (m / (v / (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.3e-32:
		tmp = -1.0 + (m / v)
	else:
		tmp = (1.0 - m) * (m / (v / (1.0 - m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.3e-32)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m / Float64(v / Float64(1.0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.3e-32)
		tmp = -1.0 + (m / v);
	else
		tmp = (1.0 - m) * (m / (v / (1.0 - m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.3e-32], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.2999999999999999e-32

   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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 99.8%

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

   5. Taylor expanded in v around 0 100.0%

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

   if 4.2999999999999999e-32 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. flip-- 99.9%

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

      3. associate-*r/ 98.0%

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

      4. associate-/r/ 98.0%

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

      5. *-commutative 98.0%

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

      6. fma-def 98.0%

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

      7. metadata-eval 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in v around 0 99.9%

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

      1. associate-/r/ 99.9%

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

      2. metadata-eval 99.9%

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

      3. +-commutative 99.9%

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

      4. flip-- 99.9%

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

      5. associate-/l* 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.3 \cdot 10^{-32}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 3: 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* 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. clear-num 99.8%

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

   2. associate-/r/ 99.8%

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

   3. clear-num 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (+ -1.0 (/ m v)) (* (+ m -2.0) (* m (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.4d0) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = (m + (-2.0d0)) * (m * (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m + -2.0) * (m * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = -1.0 + (m / v)
	else:
		tmp = (m + -2.0) * (m * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(Float64(m + -2.0) * Float64(m * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = -1.0 + (m / v);
	else
		tmp = (m + -2.0) * (m * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m + -2.0), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 98.4%

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

   5. Taylor expanded in v around 0 98.6%

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

   if 2.39999999999999991 < m

   1. Initial program 99.9%

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

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. flip-- 99.9%

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

      3. associate-*r/ 97.8%

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

      4. associate-/r/ 97.8%

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

      5. *-commutative 97.8%

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

      6. fma-def 97.8%

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

      7. metadata-eval 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf 20.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   9. Step-by-step derivation

      1. unpow2 20.0%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. associate-/l* 20.0%

         \[\leadsto -2 \cdot \color{blue}{\frac{m}{\frac{v}{m}}} + \frac{{m}^{3}}{v} \]

      3. cube-mult 20.0%

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

      4. associate-*r/ 20.0%

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

      5. associate-/l* 20.0%

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

      6. distribute-rgt-out 98.9%

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

      7. associate-/l* 98.9%

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

      8. associate-*r/ 98.9%

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

   10. Simplified 98.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 5: 98.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1.6499999999999999

   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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 98.6%

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

   if 1.6499999999999999 < m

   1. Initial program 99.9%

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

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. flip-- 99.9%

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

      3. associate-*r/ 97.8%

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

      4. associate-/r/ 97.8%

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

      5. *-commutative 97.8%

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

      6. fma-def 97.8%

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

      7. metadata-eval 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf 20.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   9. Step-by-step derivation

      1. unpow2 20.0%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. associate-/l* 20.0%

         \[\leadsto -2 \cdot \color{blue}{\frac{m}{\frac{v}{m}}} + \frac{{m}^{3}}{v} \]

      3. cube-mult 20.0%

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

      4. associate-*r/ 20.0%

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

      5. associate-/l* 20.0%

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

      6. distribute-rgt-out 98.9%

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

      7. associate-/l* 98.9%

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

      8. associate-*r/ 98.9%

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

   10. Simplified 98.9%

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

4. Final simplification 98.8%

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

Alternative 6: 98.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1.6499999999999999

   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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 98.6%

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

   if 1.6499999999999999 < m

   1. Initial program 99.9%

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

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. flip-- 99.9%

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

      3. associate-*r/ 97.8%

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

      4. associate-/r/ 97.8%

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

      5. *-commutative 97.8%

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

      6. fma-def 97.8%

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

      7. metadata-eval 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf 20.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   9. Step-by-step derivation

      1. unpow2 20.0%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. associate-/l* 20.0%

         \[\leadsto -2 \cdot \color{blue}{\frac{m}{\frac{v}{m}}} + \frac{{m}^{3}}{v} \]

      3. cube-mult 20.0%

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

      4. associate-*r/ 20.0%

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

      5. associate-/l* 20.0%

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

      6. distribute-rgt-out 98.9%

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

      7. associate-/l* 98.9%

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

      8. associate-*r/ 98.9%

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

   10. Simplified 98.9%

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

       1. associate-*r/ 98.9%

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

   12. Applied egg-rr 98.9%

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

   13. Taylor expanded in m around 0 98.9%

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

       1. unpow2 98.9%

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

       2. associate-/l* 98.9%

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

   15. Simplified 98.9%

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

4. Final simplification 98.8%

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

Alternative 7: 81.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(-m\right)}{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.5e+152) (+ -1.0 (+ m (/ m v))) (/ (* m (- m)) -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.5e+152) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * -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 <= 3.5d+152) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (m * -m) / (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.5e+152) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (m * -m) / -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.5e+152:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (m * -m) / -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.5e+152)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(-m)) / -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.5e+152)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (m * -m) / -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.5e+152], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * (-m)), $MachinePrecision] / -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.49999999999999981e152

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 78.2%

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

      1. sub-neg 78.2%

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

      2. metadata-eval 78.2%

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

      3. +-commutative 78.2%

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

      4. distribute-rgt-in 78.2%

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

      5. *-lft-identity 78.2%

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

      6. associate-*l/ 78.3%

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

      7. *-lft-identity 78.3%

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

   6. Simplified 78.3%

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

   if 3.49999999999999981e152 < m

   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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. flip-- 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-/r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 97.4%

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

   9. Taylor expanded in m around inf 97.4%

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

       1. unpow2 97.4%

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

       2. mul-1-neg 97.4%

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

       3. distribute-rgt-neg-in 97.4%

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

   11. Simplified 97.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(-m\right)}{-1}\\ \end{array} \]

Alternative 8: 81.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m \cdot m}{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.5e+152) (+ -1.0 (+ m (/ m v))) (/ (- 1.0 (* m m)) -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.5e+152) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (1.0 - (m * 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 <= 3.5d+152) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (1.0d0 - (m * m)) / (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.5e+152) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (1.0 - (m * m)) / -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.5e+152:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (1.0 - (m * m)) / -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.5e+152)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - Float64(m * m)) / -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.5e+152)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (1.0 - (m * m)) / -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.5e+152], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(m * m), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.49999999999999981e152

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 78.2%

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

      1. sub-neg 78.2%

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

      2. metadata-eval 78.2%

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

      3. +-commutative 78.2%

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

      4. distribute-rgt-in 78.2%

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

      5. *-lft-identity 78.2%

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

      6. associate-*l/ 78.3%

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

      7. *-lft-identity 78.3%

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

   6. Simplified 78.3%

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

   if 3.49999999999999981e152 < m

   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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. flip-- 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-/r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 97.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m \cdot m}{-1}\\ \end{array} \]

Alternative 9: 81.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(-m\right)}{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.5e+152) (+ -1.0 (/ m v)) (/ (* m (- m)) -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.5e+152) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m * -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 <= 3.5d+152) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = (m * -m) / (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.5e+152) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = (m * -m) / -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.5e+152:
		tmp = -1.0 + (m / v)
	else:
		tmp = (m * -m) / -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.5e+152)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(Float64(m * Float64(-m)) / -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.5e+152)
		tmp = -1.0 + (m / v);
	else
		tmp = (m * -m) / -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.5e+152], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m * (-m)), $MachinePrecision] / -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.49999999999999981e152

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 78.2%

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

   5. Taylor expanded in v around 0 78.3%

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

   if 3.49999999999999981e152 < m

   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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. flip-- 100.0%

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

      3. associate-*r/ 100.0%

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

      4. associate-/r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 97.4%

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

   9. Taylor expanded in m around inf 97.4%

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

       1. unpow2 97.4%

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

       2. mul-1-neg 97.4%

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

       3. distribute-rgt-neg-in 97.4%

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

   11. Simplified 97.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(-m\right)}{-1}\\ \end{array} \]

Alternative 10: 61.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-139}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 7e-139) -1.0 (/ m v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7e-139) {
		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 <= 7d-139) 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 <= 7e-139) {
		tmp = -1.0;
	} else {
		tmp = m / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7e-139:
		tmp = -1.0
	else:
		tmp = m / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7e-139)
		tmp = -1.0;
	else
		tmp = Float64(m / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7e-139)
		tmp = -1.0;
	else
		tmp = m / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7e-139], -1.0, N[(m / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.00000000000000002e-139

   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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 80.3%

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

   if 7.00000000000000002e-139 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. flip-- 99.9%

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

      3. associate-*r/ 98.5%

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

      4. associate-/r/ 98.5%

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

      5. *-commutative 98.5%

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

      6. fma-def 98.5%

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

      7. metadata-eval 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in v around 0 93.5%

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

      1. associate-/r/ 93.6%

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

      2. metadata-eval 93.6%

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

      3. +-commutative 93.6%

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

      4. flip-- 93.6%

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

      5. associate-/l* 93.6%

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

   10. Applied egg-rr 93.6%

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

   11. Taylor expanded in m around 0 60.7%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-139}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 11: 75.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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* 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in m around 0 75.7%

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

5. Taylor expanded in v around 0 75.8%

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

6. Final simplification 75.8%

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

Alternative 12: 27.2% 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* 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around inf 28.3%

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

   1. neg-mul-1 28.3%

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

   2. neg-sub0 28.3%

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

   3. associate--r- 28.3%

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

   4. metadata-eval 28.3%

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

6. Simplified 28.3%

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

7. Final simplification 28.3%

   \[\leadsto m + -1 \]

Alternative 13: 24.8% 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* 100.0%

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

   4. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in m around 0 25.9%

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

5. Final simplification 25.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023278 
(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)))