Rosa's DopplerBench

Percentage Accurate: 72.7% → 98.1%

Time: 16.7s

Alternatives: 17

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

C

double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 * v) / ((t1 + u) * (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Python

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

C

double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 * v) / ((t1 + u) * (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Python

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{t1}{\left(-t1\right) - u} \cdot \frac{v}{t1 + u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (* (/ t1 (- (- t1) u)) (/ v (+ t1 u))))

C

double code(double u, double v, double t1) {
	return (t1 / (-t1 - u)) * (v / (t1 + u));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (t1 / (-t1 - u)) * (v / (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return (t1 / (-t1 - u)) * (v / (t1 + u));
}

Python

def code(u, v, t1):
	return (t1 / (-t1 - u)) * (v / (t1 + u))

Julia

function code(u, v, t1)
	return Float64(Float64(t1 / Float64(Float64(-t1) - u)) * Float64(v / Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (t1 / (-t1 - u)) * (v / (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(N[(t1 / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 97.6%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   2. distribute-frac-neg 97.6%

      \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

   3. distribute-neg-frac2 97.6%

      \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   4. +-commutative 97.6%

      \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

   5. distribute-neg-in 97.6%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

   6. unsub-neg 97.6%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

3. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing

5. Final simplification 97.6%

   \[\leadsto \frac{t1}{\left(-t1\right) - u} \cdot \frac{v}{t1 + u} \]
6. Add Preprocessing

Alternative 2: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u}\\ \mathbf{if}\;t1 \leq -3.8 \cdot 10^{+69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{+139}:\\ \;\;\;\;t1 \cdot \frac{t\_1}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{u - t1}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) u))))
   (if (<= t1 -3.8e+69)
     t_1
     (if (<= t1 2.25e+139)
       (* t1 (/ t_1 (+ t1 u)))
       (* (/ v (+ t1 u)) (/ (- u t1) t1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double tmp;
	if (t1 <= -3.8e+69) {
		tmp = t_1;
	} else if (t1 <= 2.25e+139) {
		tmp = t1 * (t_1 / (t1 + u));
	} else {
		tmp = (v / (t1 + u)) * ((u - t1) / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (-t1 - u)
    if (t1 <= (-3.8d+69)) then
        tmp = t_1
    else if (t1 <= 2.25d+139) then
        tmp = t1 * (t_1 / (t1 + u))
    else
        tmp = (v / (t1 + u)) * ((u - t1) / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - u);
	double tmp;
	if (t1 <= -3.8e+69) {
		tmp = t_1;
	} else if (t1 <= 2.25e+139) {
		tmp = t1 * (t_1 / (t1 + u));
	} else {
		tmp = (v / (t1 + u)) * ((u - t1) / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-t1 - u)
	tmp = 0
	if t1 <= -3.8e+69:
		tmp = t_1
	elif t1 <= 2.25e+139:
		tmp = t1 * (t_1 / (t1 + u))
	else:
		tmp = (v / (t1 + u)) * ((u - t1) / t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - u))
	tmp = 0.0
	if (t1 <= -3.8e+69)
		tmp = t_1;
	elseif (t1 <= 2.25e+139)
		tmp = Float64(t1 * Float64(t_1 / Float64(t1 + u)));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(u - t1) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-t1 - u);
	tmp = 0.0;
	if (t1 <= -3.8e+69)
		tmp = t_1;
	elseif (t1 <= 2.25e+139)
		tmp = t1 * (t_1 / (t1 + u));
	else
		tmp = (v / (t1 + u)) * ((u - t1) / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.8e+69], t$95$1, If[LessEqual[t1, 2.25e+139], N[(t1 * N[(t$95$1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.80000000000000028e69

   1. Initial program 56.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 53.7%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 53.7%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 53.7%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 61.7%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 61.7%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. neg-mul-1 100.0%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{-1 \cdot \left(t1 + u\right)}} \]

      3. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   7. Taylor expanded in t1 around inf 90.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 90.9%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   9. Simplified 90.9%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -3.80000000000000028e69 < t1 < 2.25e139

   1. Initial program 82.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 87.1%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 87.1%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 87.1%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 92.2%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 92.2%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing

   if 2.25e139 < t1

   1. Initial program 50.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.9%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.9%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 89.7%

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{t1 + u} \]

   6. Taylor expanded in t1 around 0 89.7%

      \[\leadsto \color{blue}{\frac{u + -1 \cdot t1}{t1}} \cdot \frac{v}{t1 + u} \]
   7. Step-by-step derivation

      1. mul-1-neg 89.7%

         \[\leadsto \frac{u + \color{blue}{\left(-t1\right)}}{t1} \cdot \frac{v}{t1 + u} \]

      2. sub-neg 89.7%

         \[\leadsto \frac{\color{blue}{u - t1}}{t1} \cdot \frac{v}{t1 + u} \]

   8. Simplified 89.7%

      \[\leadsto \color{blue}{\frac{u - t1}{t1}} \cdot \frac{v}{t1 + u} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{+139}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{\left(-t1\right) - u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{u - t1}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -20000:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 3.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{u - t1}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -20000.0)
   (/ v (- (- t1) u))
   (if (<= t1 3.6e+21)
     (* (/ t1 (- u)) (/ v u))
     (* (/ v (+ t1 u)) (/ (- u t1) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -20000.0) {
		tmp = v / (-t1 - u);
	} else if (t1 <= 3.6e+21) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = (v / (t1 + u)) * ((u - t1) / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-20000.0d0)) then
        tmp = v / (-t1 - u)
    else if (t1 <= 3.6d+21) then
        tmp = (t1 / -u) * (v / u)
    else
        tmp = (v / (t1 + u)) * ((u - t1) / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -20000.0) {
		tmp = v / (-t1 - u);
	} else if (t1 <= 3.6e+21) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = (v / (t1 + u)) * ((u - t1) / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -20000.0:
		tmp = v / (-t1 - u)
	elif t1 <= 3.6e+21:
		tmp = (t1 / -u) * (v / u)
	else:
		tmp = (v / (t1 + u)) * ((u - t1) / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -20000.0)
		tmp = Float64(v / Float64(Float64(-t1) - u));
	elseif (t1 <= 3.6e+21)
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(u - t1) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -20000.0)
		tmp = v / (-t1 - u);
	elseif (t1 <= 3.6e+21)
		tmp = (t1 / -u) * (v / u);
	else
		tmp = (v / (t1 + u)) * ((u - t1) / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -20000.0], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.6e+21], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2e4

   1. Initial program 62.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 59.5%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 59.5%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 59.5%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 67.5%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 67.5%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 67.5%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. neg-mul-1 100.0%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{-1 \cdot \left(t1 + u\right)}} \]

      3. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   7. Taylor expanded in t1 around inf 89.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 89.6%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   9. Simplified 89.6%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -2e4 < t1 < 3.6e21

   1. Initial program 85.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 95.6%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 95.6%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 95.6%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 95.6%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 95.6%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 95.6%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 78.1%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{u}} \]

   6. Taylor expanded in t1 around 0 79.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{t1}{u}\right)} \cdot \frac{v}{u} \]
   7. Step-by-step derivation

      1. associate-*r/ 79.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot t1}{u}} \cdot \frac{v}{u} \]

      2. mul-1-neg 79.1%

         \[\leadsto \frac{\color{blue}{-t1}}{u} \cdot \frac{v}{u} \]

   8. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{-t1}{u}} \cdot \frac{v}{u} \]

   if 3.6e21 < t1

   1. Initial program 57.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.8%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.8%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.8%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 84.4%

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{t1 + u} \]

   6. Taylor expanded in t1 around 0 84.4%

      \[\leadsto \color{blue}{\frac{u + -1 \cdot t1}{t1}} \cdot \frac{v}{t1 + u} \]
   7. Step-by-step derivation

      1. mul-1-neg 84.4%

         \[\leadsto \frac{u + \color{blue}{\left(-t1\right)}}{t1} \cdot \frac{v}{t1 + u} \]

      2. sub-neg 84.4%

         \[\leadsto \frac{\color{blue}{u - t1}}{t1} \cdot \frac{v}{t1 + u} \]

   8. Simplified 84.4%

      \[\leadsto \color{blue}{\frac{u - t1}{t1}} \cdot \frac{v}{t1 + u} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -20000:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 3.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{u - t1}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -56000000:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -56000000.0)
   (/ v (- (- t1) u))
   (if (<= t1 6.5e+21)
     (* (/ t1 (- u)) (/ v u))
     (* (/ v (+ t1 u)) (+ (/ u t1) -1.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -56000000.0) {
		tmp = v / (-t1 - u);
	} else if (t1 <= 6.5e+21) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-56000000.0d0)) then
        tmp = v / (-t1 - u)
    else if (t1 <= 6.5d+21) then
        tmp = (t1 / -u) * (v / u)
    else
        tmp = (v / (t1 + u)) * ((u / t1) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -56000000.0) {
		tmp = v / (-t1 - u);
	} else if (t1 <= 6.5e+21) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -56000000.0:
		tmp = v / (-t1 - u)
	elif t1 <= 6.5e+21:
		tmp = (t1 / -u) * (v / u)
	else:
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -56000000.0)
		tmp = Float64(v / Float64(Float64(-t1) - u));
	elseif (t1 <= 6.5e+21)
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(u / t1) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -56000000.0)
		tmp = v / (-t1 - u);
	elseif (t1 <= 6.5e+21)
		tmp = (t1 / -u) * (v / u);
	else
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -56000000.0], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6.5e+21], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.6e7

   1. Initial program 62.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 59.5%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 59.5%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 59.5%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 67.5%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 67.5%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 67.5%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. neg-mul-1 100.0%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{-1 \cdot \left(t1 + u\right)}} \]

      3. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   7. Taylor expanded in t1 around inf 89.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 89.6%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   9. Simplified 89.6%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -5.6e7 < t1 < 6.5e21

   1. Initial program 85.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 95.6%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 95.6%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 95.6%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 95.6%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 95.6%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 95.6%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 78.1%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{u}} \]

   6. Taylor expanded in t1 around 0 79.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{t1}{u}\right)} \cdot \frac{v}{u} \]
   7. Step-by-step derivation

      1. associate-*r/ 79.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot t1}{u}} \cdot \frac{v}{u} \]

      2. mul-1-neg 79.1%

         \[\leadsto \frac{\color{blue}{-t1}}{u} \cdot \frac{v}{u} \]

   8. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{-t1}{u}} \cdot \frac{v}{u} \]

   if 6.5e21 < t1

   1. Initial program 57.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.8%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.8%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.8%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 84.4%

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{t1 + u} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -56000000:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -25000:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 + u}{v}}}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -25000.0)
   (/ v (- (- t1) u))
   (if (<= t1 5.8e-116)
     (* (/ t1 (- u)) (/ v u))
     (/ (/ t1 (/ (+ t1 u) v)) (- t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -25000.0) {
		tmp = v / (-t1 - u);
	} else if (t1 <= 5.8e-116) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = (t1 / ((t1 + u) / v)) / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-25000.0d0)) then
        tmp = v / (-t1 - u)
    else if (t1 <= 5.8d-116) then
        tmp = (t1 / -u) * (v / u)
    else
        tmp = (t1 / ((t1 + u) / v)) / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -25000.0) {
		tmp = v / (-t1 - u);
	} else if (t1 <= 5.8e-116) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = (t1 / ((t1 + u) / v)) / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -25000.0:
		tmp = v / (-t1 - u)
	elif t1 <= 5.8e-116:
		tmp = (t1 / -u) * (v / u)
	else:
		tmp = (t1 / ((t1 + u) / v)) / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -25000.0)
		tmp = Float64(v / Float64(Float64(-t1) - u));
	elseif (t1 <= 5.8e-116)
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	else
		tmp = Float64(Float64(t1 / Float64(Float64(t1 + u) / v)) / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -25000.0)
		tmp = v / (-t1 - u);
	elseif (t1 <= 5.8e-116)
		tmp = (t1 / -u) * (v / u);
	else
		tmp = (t1 / ((t1 + u) / v)) / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -25000.0], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.8e-116], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] / (-t1)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -25000

   1. Initial program 62.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 59.5%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 59.5%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 59.5%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 67.5%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 67.5%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 67.5%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. neg-mul-1 100.0%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{-1 \cdot \left(t1 + u\right)}} \]

      3. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   7. Taylor expanded in t1 around inf 89.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 89.6%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   9. Simplified 89.6%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -25000 < t1 < 5.7999999999999996e-116

   1. Initial program 82.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 94.7%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 94.7%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 94.7%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 94.7%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 94.7%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 94.7%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 82.7%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{u}} \]

   6. Taylor expanded in t1 around 0 82.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{t1}{u}\right)} \cdot \frac{v}{u} \]
   7. Step-by-step derivation

      1. associate-*r/ 82.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot t1}{u}} \cdot \frac{v}{u} \]

      2. mul-1-neg 82.9%

         \[\leadsto \frac{\color{blue}{-t1}}{u} \cdot \frac{v}{u} \]

   8. Simplified 82.9%

      \[\leadsto \color{blue}{\frac{-t1}{u}} \cdot \frac{v}{u} \]

   if 5.7999999999999996e-116 < t1

   1. Initial program 69.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 76.7%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 76.7%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 76.7%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 87.2%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 87.2%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. neg-mul-1 99.9%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{-1 \cdot \left(t1 + u\right)}} \]

      3. associate-/r* 99.9%

         \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]
   7. Step-by-step derivation

      1. frac-2neg 99.9%

         \[\leadsto \frac{\color{blue}{\frac{-t1 \cdot \frac{v}{t1 + u}}{--1}}}{t1 + u} \]

      2. metadata-eval 99.9%

         \[\leadsto \frac{\frac{-t1 \cdot \frac{v}{t1 + u}}{\color{blue}{1}}}{t1 + u} \]

      3. /-rgt-identity 99.9%

         \[\leadsto \frac{\color{blue}{-t1 \cdot \frac{v}{t1 + u}}}{t1 + u} \]

      4. clear-num 99.2%

         \[\leadsto \frac{-t1 \cdot \color{blue}{\frac{1}{\frac{t1 + u}{v}}}}{t1 + u} \]

      5. un-div-inv 99.4%

         \[\leadsto \frac{-\color{blue}{\frac{t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{\color{blue}{-\frac{t1}{\frac{t1 + u}{v}}}}{t1 + u} \]

   9. Taylor expanded in t1 around inf 77.8%

      \[\leadsto \frac{-\frac{t1}{\frac{t1 + u}{v}}}{\color{blue}{t1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -25000:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 + u}{v}}}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -27000 \lor \neg \left(t1 \leq 1.45 \cdot 10^{-103}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -27000.0) (not (<= t1 1.45e-103)))
   (/ v (- (- t1) u))
   (* (/ t1 (- u)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -27000.0) || !(t1 <= 1.45e-103)) {
		tmp = v / (-t1 - u);
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-27000.0d0)) .or. (.not. (t1 <= 1.45d-103))) then
        tmp = v / (-t1 - u)
    else
        tmp = (t1 / -u) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -27000.0) || !(t1 <= 1.45e-103)) {
		tmp = v / (-t1 - u);
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -27000.0) or not (t1 <= 1.45e-103):
		tmp = v / (-t1 - u)
	else:
		tmp = (t1 / -u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -27000.0) || !(t1 <= 1.45e-103))
		tmp = Float64(v / Float64(Float64(-t1) - u));
	else
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -27000.0) || ~((t1 <= 1.45e-103)))
		tmp = v / (-t1 - u);
	else
		tmp = (t1 / -u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -27000.0], N[Not[LessEqual[t1, 1.45e-103]], $MachinePrecision]], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -27000 or 1.4499999999999999e-103 < t1

   1. Initial program 66.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 69.3%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 69.3%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 69.3%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 78.8%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 78.8%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. neg-mul-1 99.9%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{-1 \cdot \left(t1 + u\right)}} \]

      3. associate-/r* 99.9%

         \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   7. Taylor expanded in t1 around inf 82.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 82.2%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   9. Simplified 82.2%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -27000 < t1 < 1.4499999999999999e-103

   1. Initial program 83.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 94.7%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 94.7%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 94.7%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 94.7%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 94.7%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 94.7%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 82.8%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{u}} \]

   6. Taylor expanded in t1 around 0 83.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{t1}{u}\right)} \cdot \frac{v}{u} \]
   7. Step-by-step derivation

      1. associate-*r/ 83.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot t1}{u}} \cdot \frac{v}{u} \]

      2. mul-1-neg 83.1%

         \[\leadsto \frac{\color{blue}{-t1}}{u} \cdot \frac{v}{u} \]

   8. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{-t1}{u}} \cdot \frac{v}{u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -27000 \lor \neg \left(t1 \leq 1.45 \cdot 10^{-103}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -150000 \lor \neg \left(t1 \leq 1.45 \cdot 10^{-103}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -150000.0) (not (<= t1 1.45e-103)))
   (/ v (- (- t1) u))
   (* t1 (/ (/ (- v) u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -150000.0) || !(t1 <= 1.45e-103)) {
		tmp = v / (-t1 - u);
	} else {
		tmp = t1 * ((-v / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-150000.0d0)) .or. (.not. (t1 <= 1.45d-103))) then
        tmp = v / (-t1 - u)
    else
        tmp = t1 * ((-v / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -150000.0) || !(t1 <= 1.45e-103)) {
		tmp = v / (-t1 - u);
	} else {
		tmp = t1 * ((-v / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -150000.0) or not (t1 <= 1.45e-103):
		tmp = v / (-t1 - u)
	else:
		tmp = t1 * ((-v / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -150000.0) || !(t1 <= 1.45e-103))
		tmp = Float64(v / Float64(Float64(-t1) - u));
	else
		tmp = Float64(t1 * Float64(Float64(Float64(-v) / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -150000.0) || ~((t1 <= 1.45e-103)))
		tmp = v / (-t1 - u);
	else
		tmp = t1 * ((-v / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -150000.0], N[Not[LessEqual[t1, 1.45e-103]], $MachinePrecision]], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[((-v) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.5e5 or 1.4499999999999999e-103 < t1

   1. Initial program 66.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 69.3%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 69.3%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 69.3%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 78.8%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 78.8%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. neg-mul-1 99.9%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{-1 \cdot \left(t1 + u\right)}} \]

      3. associate-/r* 99.9%

         \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   7. Taylor expanded in t1 around inf 82.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 82.2%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   9. Simplified 82.2%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -1.5e5 < t1 < 1.4499999999999999e-103

   1. Initial program 83.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 85.2%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 85.2%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 85.2%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 91.4%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 91.4%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 81.8%

      \[\leadsto t1 \cdot \frac{\color{blue}{\frac{v}{u}}}{-\left(t1 + u\right)} \]

   6. Taylor expanded in t1 around 0 82.0%

      \[\leadsto t1 \cdot \frac{\frac{v}{u}}{\color{blue}{-1 \cdot u}} \]
   7. Step-by-step derivation

      1. neg-mul-1 82.0%

         \[\leadsto t1 \cdot \frac{\frac{v}{u}}{\color{blue}{-u}} \]

   8. Simplified 82.0%

      \[\leadsto t1 \cdot \frac{\frac{v}{u}}{\color{blue}{-u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -150000 \lor \neg \left(t1 \leq 1.45 \cdot 10^{-103}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{-141} \lor \neg \left(t1 \leq 2.2 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.5e-141) (not (<= t1 2.2e-111)))
   (/ v (- (- t1) u))
   (/ v (* u (/ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e-141) || !(t1 <= 2.2e-111)) {
		tmp = v / (-t1 - u);
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4.5d-141)) .or. (.not. (t1 <= 2.2d-111))) then
        tmp = v / (-t1 - u)
    else
        tmp = v / (u * (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e-141) || !(t1 <= 2.2e-111)) {
		tmp = v / (-t1 - u);
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.5e-141) or not (t1 <= 2.2e-111):
		tmp = v / (-t1 - u)
	else:
		tmp = v / (u * (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.5e-141) || !(t1 <= 2.2e-111))
		tmp = Float64(v / Float64(Float64(-t1) - u));
	else
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.5e-141) || ~((t1 <= 2.2e-111)))
		tmp = v / (-t1 - u);
	else
		tmp = v / (u * (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.5e-141], N[Not[LessEqual[t1, 2.2e-111]], $MachinePrecision]], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.5e-141 or 2.2e-111 < t1

   1. Initial program 70.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 71.4%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 71.4%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 71.4%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 80.4%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 80.4%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. neg-mul-1 99.9%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{-1 \cdot \left(t1 + u\right)}} \]

      3. associate-/r* 99.9%

         \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   7. Taylor expanded in t1 around inf 77.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 77.2%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   9. Simplified 77.2%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -4.5e-141 < t1 < 2.2e-111

   1. Initial program 79.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 93.1%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 93.1%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 93.1%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 93.1%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 93.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 93.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 86.3%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{u}} \]

   6. Taylor expanded in t1 around 0 86.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{t1}{u}\right)} \cdot \frac{v}{u} \]
   7. Step-by-step derivation

      1. associate-*r/ 86.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot t1}{u}} \cdot \frac{v}{u} \]

      2. mul-1-neg 86.3%

         \[\leadsto \frac{\color{blue}{-t1}}{u} \cdot \frac{v}{u} \]

   8. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{-t1}{u}} \cdot \frac{v}{u} \]
   9. Step-by-step derivation

      1. clear-num 85.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{u}{-t1}}} \cdot \frac{v}{u} \]

      2. frac-times 84.1%

         \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{u}{-t1} \cdot u}} \]

      3. *-un-lft-identity 84.1%

         \[\leadsto \frac{\color{blue}{v}}{\frac{u}{-t1} \cdot u} \]

      4. add-sqr-sqrt 34.4%

         \[\leadsto \frac{v}{\frac{u}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}} \cdot u} \]

      5. sqrt-unprod 47.4%

         \[\leadsto \frac{v}{\frac{u}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}} \cdot u} \]

      6. sqr-neg 47.4%

         \[\leadsto \frac{v}{\frac{u}{\sqrt{\color{blue}{t1 \cdot t1}}} \cdot u} \]

      7. sqrt-unprod 27.1%

         \[\leadsto \frac{v}{\frac{u}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}} \cdot u} \]

      8. add-sqr-sqrt 45.6%

         \[\leadsto \frac{v}{\frac{u}{\color{blue}{t1}} \cdot u} \]

   10. Applied egg-rr 45.6%

       \[\leadsto \color{blue}{\frac{v}{\frac{u}{t1} \cdot u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{-141} \lor \neg \left(t1 \leq 2.2 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{-142} \lor \neg \left(t1 \leq 2.1 \cdot 10^{-112}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.2e-142) (not (<= t1 2.1e-112)))
   (/ v (- (- t1) u))
   (/ t1 (* u (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.2e-142) || !(t1 <= 2.1e-112)) {
		tmp = v / (-t1 - u);
	} else {
		tmp = t1 / (u * (u / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-9.2d-142)) .or. (.not. (t1 <= 2.1d-112))) then
        tmp = v / (-t1 - u)
    else
        tmp = t1 / (u * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.2e-142) || !(t1 <= 2.1e-112)) {
		tmp = v / (-t1 - u);
	} else {
		tmp = t1 / (u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9.2e-142) or not (t1 <= 2.1e-112):
		tmp = v / (-t1 - u)
	else:
		tmp = t1 / (u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9.2e-142) || !(t1 <= 2.1e-112))
		tmp = Float64(v / Float64(Float64(-t1) - u));
	else
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -9.2e-142) || ~((t1 <= 2.1e-112)))
		tmp = v / (-t1 - u);
	else
		tmp = t1 / (u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9.2e-142], N[Not[LessEqual[t1, 2.1e-112]], $MachinePrecision]], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.20000000000000009e-142 or 2.1000000000000001e-112 < t1

   1. Initial program 70.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 71.4%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 71.4%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 71.4%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 80.4%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 80.4%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. neg-mul-1 99.9%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{-1 \cdot \left(t1 + u\right)}} \]

      3. associate-/r* 99.9%

         \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

   7. Taylor expanded in t1 around inf 77.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 77.2%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   9. Simplified 77.2%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -9.20000000000000009e-142 < t1 < 2.1000000000000001e-112

   1. Initial program 79.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 93.1%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 93.1%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 93.1%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 93.1%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 93.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 93.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 93.1%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 86.3%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{u}} \]

   6. Taylor expanded in t1 around 0 86.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{t1}{u}\right)} \cdot \frac{v}{u} \]
   7. Step-by-step derivation

      1. associate-*r/ 86.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot t1}{u}} \cdot \frac{v}{u} \]

      2. mul-1-neg 86.3%

         \[\leadsto \frac{\color{blue}{-t1}}{u} \cdot \frac{v}{u} \]

   8. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{-t1}{u}} \cdot \frac{v}{u} \]
   9. Step-by-step derivation

      1. *-commutative 86.3%

         \[\leadsto \color{blue}{\frac{v}{u} \cdot \frac{-t1}{u}} \]

      2. clear-num 86.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \cdot \frac{-t1}{u} \]

      3. frac-times 86.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(-t1\right)}{\frac{u}{v} \cdot u}} \]

      4. *-un-lft-identity 86.9%

         \[\leadsto \frac{\color{blue}{-t1}}{\frac{u}{v} \cdot u} \]

      5. add-sqr-sqrt 38.7%

         \[\leadsto \frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{\frac{u}{v} \cdot u} \]

      6. sqrt-unprod 47.1%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{\frac{u}{v} \cdot u} \]

      7. sqr-neg 47.1%

         \[\leadsto \frac{\sqrt{\color{blue}{t1 \cdot t1}}}{\frac{u}{v} \cdot u} \]

      8. sqrt-unprod 27.0%

         \[\leadsto \frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{\frac{u}{v} \cdot u} \]

      9. add-sqr-sqrt 45.4%

         \[\leadsto \frac{\color{blue}{t1}}{\frac{u}{v} \cdot u} \]

   10. Applied egg-rr 45.4%

       \[\leadsto \color{blue}{\frac{t1}{\frac{u}{v} \cdot u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{-142} \lor \neg \left(t1 \leq 2.1 \cdot 10^{-112}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+186} \lor \neg \left(u \leq 1.1 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.6e+186) (not (<= u 1.1e+161))) (/ 1.0 (/ u v)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.6e+186) || !(u <= 1.1e+161)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.6d+186)) .or. (.not. (u <= 1.1d+161))) then
        tmp = 1.0d0 / (u / v)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.6e+186) || !(u <= 1.1e+161)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.6e+186) or not (u <= 1.1e+161):
		tmp = 1.0 / (u / v)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.6e+186) || !(u <= 1.1e+161))
		tmp = Float64(1.0 / Float64(u / v));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.6e+186) || ~((u <= 1.1e+161)))
		tmp = 1.0 / (u / v);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.6e+186], N[Not[LessEqual[u, 1.1e+161]], $MachinePrecision]], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.6e186 or 1.1e161 < u

   1. Initial program 81.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf 58.2%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]
   4. Step-by-step derivation

      1. associate-/l* 56.8%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot t1}} \]

      2. add-sqr-sqrt 26.0%

         \[\leadsto \color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      3. sqrt-unprod 57.3%

         \[\leadsto \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      4. sqr-neg 57.3%

         \[\leadsto \sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      5. sqrt-unprod 30.9%

         \[\leadsto \color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      6. add-sqr-sqrt 57.0%

         \[\leadsto \color{blue}{t1} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      7. *-commutative 57.0%

         \[\leadsto t1 \cdot \frac{v}{\color{blue}{t1 \cdot \left(t1 + u\right)}} \]

   5. Applied egg-rr 57.0%

      \[\leadsto \color{blue}{t1 \cdot \frac{v}{t1 \cdot \left(t1 + u\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 58.3%

         \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 \cdot \left(t1 + u\right)}} \]

      2. *-commutative 58.3%

         \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 \cdot \left(t1 + u\right)} \]

      3. associate-*r/ 57.0%

         \[\leadsto \color{blue}{v \cdot \frac{t1}{t1 \cdot \left(t1 + u\right)}} \]

      4. associate-/r* 56.8%

         \[\leadsto v \cdot \color{blue}{\frac{\frac{t1}{t1}}{t1 + u}} \]

      5. *-inverses 56.8%

         \[\leadsto v \cdot \frac{\color{blue}{1}}{t1 + u} \]

   7. Simplified 56.8%

      \[\leadsto \color{blue}{v \cdot \frac{1}{t1 + u}} \]

   8. Taylor expanded in t1 around 0 56.8%

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

      1. clear-num 58.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \]

      2. inv-pow 58.4%

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

   10. Applied egg-rr 58.4%

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

       1. unpow-1 58.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \]

   12. Simplified 58.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \]

   if -1.6e186 < u < 1.1e161

   1. Initial program 71.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 74.5%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 74.5%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 74.5%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 82.7%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 82.7%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 61.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 61.0%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+186} \lor \neg \left(u \leq 1.1 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+186} \lor \neg \left(u \leq 1.15 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.4e+186) (not (<= u 1.15e+161))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+186) || !(u <= 1.15e+161)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.4d+186)) .or. (.not. (u <= 1.15d+161))) then
        tmp = -v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+186) || !(u <= 1.15e+161)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.4e+186) or not (u <= 1.15e+161):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.4e+186) || !(u <= 1.15e+161))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.4e+186) || ~((u <= 1.15e+161)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.4e+186], N[Not[LessEqual[u, 1.15e+161]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.40000000000000005e186 or 1.15e161 < u

   1. Initial program 81.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 81.7%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 81.7%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 81.7%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 89.3%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 89.3%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 89.3%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 89.3%

      \[\leadsto t1 \cdot \frac{\color{blue}{\frac{v}{u}}}{-\left(t1 + u\right)} \]

   6. Taylor expanded in t1 around inf 57.1%

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

      1. neg-mul-1 57.1%

         \[\leadsto \color{blue}{-\frac{v}{u}} \]

      2. distribute-neg-frac 57.1%

         \[\leadsto \color{blue}{\frac{-v}{u}} \]

   8. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]

   if -3.40000000000000005e186 < u < 1.15e161

   1. Initial program 71.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 74.5%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 74.5%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 74.5%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 82.7%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 82.7%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 61.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 61.0%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+186} \lor \neg \left(u \leq 1.15 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{+187} \lor \neg \left(u \leq 1.15 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.5e+187) (not (<= u 1.15e+161))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.5e+187) || !(u <= 1.15e+161)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.5d+187)) .or. (.not. (u <= 1.15d+161))) then
        tmp = v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.5e+187) || !(u <= 1.15e+161)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.5e+187) or not (u <= 1.15e+161):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.5e+187) || !(u <= 1.15e+161))
		tmp = Float64(v / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.5e+187) || ~((u <= 1.15e+161)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.5e+187], N[Not[LessEqual[u, 1.15e+161]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.5e187 or 1.15e161 < u

   1. Initial program 81.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf 58.2%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]
   4. Step-by-step derivation

      1. associate-/l* 56.8%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot t1}} \]

      2. add-sqr-sqrt 26.0%

         \[\leadsto \color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      3. sqrt-unprod 57.3%

         \[\leadsto \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      4. sqr-neg 57.3%

         \[\leadsto \sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      5. sqrt-unprod 30.9%

         \[\leadsto \color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      6. add-sqr-sqrt 57.0%

         \[\leadsto \color{blue}{t1} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      7. *-commutative 57.0%

         \[\leadsto t1 \cdot \frac{v}{\color{blue}{t1 \cdot \left(t1 + u\right)}} \]

   5. Applied egg-rr 57.0%

      \[\leadsto \color{blue}{t1 \cdot \frac{v}{t1 \cdot \left(t1 + u\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 58.3%

         \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 \cdot \left(t1 + u\right)}} \]

      2. *-commutative 58.3%

         \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 \cdot \left(t1 + u\right)} \]

      3. associate-*r/ 57.0%

         \[\leadsto \color{blue}{v \cdot \frac{t1}{t1 \cdot \left(t1 + u\right)}} \]

      4. associate-/r* 56.8%

         \[\leadsto v \cdot \color{blue}{\frac{\frac{t1}{t1}}{t1 + u}} \]

      5. *-inverses 56.8%

         \[\leadsto v \cdot \frac{\color{blue}{1}}{t1 + u} \]

   7. Simplified 56.8%

      \[\leadsto \color{blue}{v \cdot \frac{1}{t1 + u}} \]

   8. Taylor expanded in t1 around 0 56.8%

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

   if -1.5e187 < u < 1.15e161

   1. Initial program 71.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 74.5%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 74.5%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 74.5%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 82.7%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 82.7%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 61.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 61.0%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{+187} \lor \neg \left(u \leq 1.15 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 22.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -350000000 \lor \neg \left(t1 \leq 9.2 \cdot 10^{+213}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -350000000.0) (not (<= t1 9.2e+213))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -350000000.0) || !(t1 <= 9.2e+213)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-350000000.0d0)) .or. (.not. (t1 <= 9.2d+213))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -350000000.0) || !(t1 <= 9.2e+213)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -350000000.0) or not (t1 <= 9.2e+213):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -350000000.0) || !(t1 <= 9.2e+213))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -350000000.0) || ~((t1 <= 9.2e+213)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -350000000.0], N[Not[LessEqual[t1, 9.2e+213]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.5e8 or 9.19999999999999992e213 < t1

   1. Initial program 59.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.9%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.9%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 89.5%

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{t1 + u} \]

   6. Taylor expanded in u around inf 34.9%

      \[\leadsto \color{blue}{\frac{v}{t1}} \]

   if -3.5e8 < t1 < 9.19999999999999992e213

   1. Initial program 79.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf 42.6%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]
   4. Step-by-step derivation

      1. associate-/l* 41.5%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot t1}} \]

      2. add-sqr-sqrt 9.8%

         \[\leadsto \color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      3. sqrt-unprod 30.0%

         \[\leadsto \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      4. sqr-neg 30.0%

         \[\leadsto \sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      5. sqrt-unprod 16.3%

         \[\leadsto \color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      6. add-sqr-sqrt 22.9%

         \[\leadsto \color{blue}{t1} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      7. *-commutative 22.9%

         \[\leadsto t1 \cdot \frac{v}{\color{blue}{t1 \cdot \left(t1 + u\right)}} \]

   5. Applied egg-rr 22.9%

      \[\leadsto \color{blue}{t1 \cdot \frac{v}{t1 \cdot \left(t1 + u\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 26.6%

         \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 \cdot \left(t1 + u\right)}} \]

      2. *-commutative 26.6%

         \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 \cdot \left(t1 + u\right)} \]

      3. associate-*r/ 23.0%

         \[\leadsto \color{blue}{v \cdot \frac{t1}{t1 \cdot \left(t1 + u\right)}} \]

      4. associate-/r* 23.0%

         \[\leadsto v \cdot \color{blue}{\frac{\frac{t1}{t1}}{t1 + u}} \]

      5. *-inverses 23.0%

         \[\leadsto v \cdot \frac{\color{blue}{1}}{t1 + u} \]

   7. Simplified 23.0%

      \[\leadsto \color{blue}{v \cdot \frac{1}{t1 + u}} \]

   8. Taylor expanded in t1 around 0 23.1%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -350000000 \lor \neg \left(t1 \leq 9.2 \cdot 10^{+213}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{t1 - u}{v}} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ -1.0 (/ (- t1 u) v)))

C

double code(double u, double v, double t1) {
	return -1.0 / ((t1 - u) / v);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-1.0d0) / ((t1 - u) / v)
end function

Java

public static double code(double u, double v, double t1) {
	return -1.0 / ((t1 - u) / v);
}

Python

def code(u, v, t1):
	return -1.0 / ((t1 - u) / v)

Julia

function code(u, v, t1)
	return Float64(-1.0 / Float64(Float64(t1 - u) / v))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = -1.0 / ((t1 - u) / v);
end

Wolfram

code[u_, v_, t1_] := N[(-1.0 / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing

3. Taylor expanded in t1 around inf 47.2%

   \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]
4. Step-by-step derivation

   1. associate-/l* 45.5%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot t1}} \]

   2. add-sqr-sqrt 20.5%

      \[\leadsto \color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

   3. sqrt-unprod 28.7%

      \[\leadsto \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

   4. sqr-neg 28.7%

      \[\leadsto \sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

   5. sqrt-unprod 14.0%

      \[\leadsto \color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

   6. add-sqr-sqrt 26.9%

      \[\leadsto \color{blue}{t1} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

   7. *-commutative 26.9%

      \[\leadsto t1 \cdot \frac{v}{\color{blue}{t1 \cdot \left(t1 + u\right)}} \]

5. Applied egg-rr 26.9%

   \[\leadsto \color{blue}{t1 \cdot \frac{v}{t1 \cdot \left(t1 + u\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 29.4%

      \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 \cdot \left(t1 + u\right)}} \]

   2. *-commutative 29.4%

      \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 \cdot \left(t1 + u\right)} \]

   3. associate-*r/ 26.9%

      \[\leadsto \color{blue}{v \cdot \frac{t1}{t1 \cdot \left(t1 + u\right)}} \]

   4. associate-/r* 26.8%

      \[\leadsto v \cdot \color{blue}{\frac{\frac{t1}{t1}}{t1 + u}} \]

   5. *-inverses 26.8%

      \[\leadsto v \cdot \frac{\color{blue}{1}}{t1 + u} \]

7. Simplified 26.8%

   \[\leadsto \color{blue}{v \cdot \frac{1}{t1 + u}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 11.8%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{t1 + u} \cdot \sqrt{t1 + u}}} \]

   2. sqrt-prod 42.7%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{\left(t1 + u\right) \cdot \left(t1 + u\right)}}} \]

   3. sqr-neg 42.7%

      \[\leadsto v \cdot \frac{1}{\sqrt{\color{blue}{\left(-\left(t1 + u\right)\right) \cdot \left(-\left(t1 + u\right)\right)}}} \]

   4. sqrt-unprod 31.1%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{-\left(t1 + u\right)} \cdot \sqrt{-\left(t1 + u\right)}}} \]

   5. add-sqr-sqrt 60.5%

      \[\leadsto v \cdot \frac{1}{\color{blue}{-\left(t1 + u\right)}} \]

   6. +-commutative 60.5%

      \[\leadsto v \cdot \frac{1}{-\color{blue}{\left(u + t1\right)}} \]

   7. distribute-neg-in 60.5%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   8. sub-neg 60.5%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\left(-u\right) - t1}} \]

   9. add-sqr-sqrt 32.0%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} - t1} \]

   10. sqrt-unprod 66.1%

       \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} - t1} \]

   11. sqr-neg 66.1%

       \[\leadsto v \cdot \frac{1}{\sqrt{\color{blue}{u \cdot u}} - t1} \]

   12. sqrt-unprod 28.1%

       \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{u} \cdot \sqrt{u}} - t1} \]

   13. add-sqr-sqrt 60.4%

       \[\leadsto v \cdot \frac{1}{\color{blue}{u} - t1} \]

9. Applied egg-rr 60.4%

   \[\leadsto v \cdot \frac{1}{\color{blue}{u - t1}} \]
10. Step-by-step derivation

    1. un-div-inv 60.6%

       \[\leadsto \color{blue}{\frac{v}{u - t1}} \]

    2. sub-neg 60.6%

       \[\leadsto \frac{v}{\color{blue}{u + \left(-t1\right)}} \]

    3. add-sqr-sqrt 29.5%

       \[\leadsto \frac{v}{u + \color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}} \]

    4. sqrt-unprod 37.3%

       \[\leadsto \frac{v}{u + \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}} \]

    5. sqr-neg 37.3%

       \[\leadsto \frac{v}{u + \sqrt{\color{blue}{t1 \cdot t1}}} \]

    6. sqrt-unprod 13.9%

       \[\leadsto \frac{v}{u + \color{blue}{\sqrt{t1} \cdot \sqrt{t1}}} \]

    7. add-sqr-sqrt 26.8%

       \[\leadsto \frac{v}{u + \color{blue}{t1}} \]

    8. +-commutative 26.8%

       \[\leadsto \frac{v}{\color{blue}{t1 + u}} \]

    9. clear-num 27.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \]

    10. frac-2neg 27.2%

        \[\leadsto \color{blue}{\frac{-1}{-\frac{t1 + u}{v}}} \]

    11. metadata-eval 27.2%

        \[\leadsto \frac{\color{blue}{-1}}{-\frac{t1 + u}{v}} \]

    12. distribute-neg-frac 27.2%

        \[\leadsto \frac{-1}{\color{blue}{\frac{-\left(t1 + u\right)}{v}}} \]

    13. distribute-neg-in 27.2%

        \[\leadsto \frac{-1}{\frac{\color{blue}{\left(-t1\right) + \left(-u\right)}}{v}} \]

    14. add-sqr-sqrt 13.1%

        \[\leadsto \frac{-1}{\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)}{v}} \]

    15. sqrt-unprod 39.6%

        \[\leadsto \frac{-1}{\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)}{v}} \]

    16. sqr-neg 39.6%

        \[\leadsto \frac{-1}{\frac{\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)}{v}} \]

    17. sqrt-unprod 30.9%

        \[\leadsto \frac{-1}{\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)}{v}} \]

    18. add-sqr-sqrt 60.7%

        \[\leadsto \frac{-1}{\frac{\color{blue}{t1} + \left(-u\right)}{v}} \]

    19. sub-neg 60.7%

        \[\leadsto \frac{-1}{\frac{\color{blue}{t1 - u}}{v}} \]

11. Applied egg-rr 60.7%

    \[\leadsto \color{blue}{\frac{-1}{\frac{t1 - u}{v}}} \]
12. Add Preprocessing

Alternative 15: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(-t1\right) - u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- (- t1) u)))

C

double code(double u, double v, double t1) {
	return v / (-t1 - u);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / (-t1 - u)
end function

Java

public static double code(double u, double v, double t1) {
	return v / (-t1 - u);
}

Python

def code(u, v, t1):
	return v / (-t1 - u)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(-t1) - u))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / (-t1 - u);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/l* 76.3%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. distribute-lft-neg-out 76.3%

      \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. distribute-rgt-neg-in 76.3%

      \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

   4. associate-/r* 84.3%

      \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

   5. distribute-neg-frac2 84.3%

      \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

3. Simplified 84.3%

   \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 97.2%

      \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   2. neg-mul-1 97.2%

      \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{-1 \cdot \left(t1 + u\right)}} \]

   3. associate-/r* 97.2%

      \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

6. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u}} \]

7. Taylor expanded in t1 around inf 60.7%

   \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
8. Step-by-step derivation

   1. mul-1-neg 60.7%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

9. Simplified 60.7%

   \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

10. Final simplification 60.7%

    \[\leadsto \frac{v}{\left(-t1\right) - u} \]
11. Add Preprocessing

Alternative 16: 61.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- u t1)))

C

double code(double u, double v, double t1) {
	return v / (u - t1);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / (u - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / (u - t1);
}

Python

def code(u, v, t1):
	return v / (u - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(u - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / (u - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing

3. Taylor expanded in t1 around inf 47.2%

   \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]
4. Step-by-step derivation

   1. associate-/l* 45.5%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot t1}} \]

   2. add-sqr-sqrt 20.5%

      \[\leadsto \color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

   3. sqrt-unprod 28.7%

      \[\leadsto \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

   4. sqr-neg 28.7%

      \[\leadsto \sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

   5. sqrt-unprod 14.0%

      \[\leadsto \color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

   6. add-sqr-sqrt 26.9%

      \[\leadsto \color{blue}{t1} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

   7. *-commutative 26.9%

      \[\leadsto t1 \cdot \frac{v}{\color{blue}{t1 \cdot \left(t1 + u\right)}} \]

5. Applied egg-rr 26.9%

   \[\leadsto \color{blue}{t1 \cdot \frac{v}{t1 \cdot \left(t1 + u\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 29.4%

      \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 \cdot \left(t1 + u\right)}} \]

   2. *-commutative 29.4%

      \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 \cdot \left(t1 + u\right)} \]

   3. associate-*r/ 26.9%

      \[\leadsto \color{blue}{v \cdot \frac{t1}{t1 \cdot \left(t1 + u\right)}} \]

   4. associate-/r* 26.8%

      \[\leadsto v \cdot \color{blue}{\frac{\frac{t1}{t1}}{t1 + u}} \]

   5. *-inverses 26.8%

      \[\leadsto v \cdot \frac{\color{blue}{1}}{t1 + u} \]

7. Simplified 26.8%

   \[\leadsto \color{blue}{v \cdot \frac{1}{t1 + u}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 11.8%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{t1 + u} \cdot \sqrt{t1 + u}}} \]

   2. sqrt-prod 42.7%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{\left(t1 + u\right) \cdot \left(t1 + u\right)}}} \]

   3. sqr-neg 42.7%

      \[\leadsto v \cdot \frac{1}{\sqrt{\color{blue}{\left(-\left(t1 + u\right)\right) \cdot \left(-\left(t1 + u\right)\right)}}} \]

   4. sqrt-unprod 31.1%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{-\left(t1 + u\right)} \cdot \sqrt{-\left(t1 + u\right)}}} \]

   5. add-sqr-sqrt 60.5%

      \[\leadsto v \cdot \frac{1}{\color{blue}{-\left(t1 + u\right)}} \]

   6. +-commutative 60.5%

      \[\leadsto v \cdot \frac{1}{-\color{blue}{\left(u + t1\right)}} \]

   7. distribute-neg-in 60.5%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   8. sub-neg 60.5%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\left(-u\right) - t1}} \]

   9. add-sqr-sqrt 32.0%

      \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} - t1} \]

   10. sqrt-unprod 66.1%

       \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} - t1} \]

   11. sqr-neg 66.1%

       \[\leadsto v \cdot \frac{1}{\sqrt{\color{blue}{u \cdot u}} - t1} \]

   12. sqrt-unprod 28.1%

       \[\leadsto v \cdot \frac{1}{\color{blue}{\sqrt{u} \cdot \sqrt{u}} - t1} \]

   13. add-sqr-sqrt 60.4%

       \[\leadsto v \cdot \frac{1}{\color{blue}{u} - t1} \]

9. Applied egg-rr 60.4%

   \[\leadsto v \cdot \frac{1}{\color{blue}{u - t1}} \]

10. Taylor expanded in v around 0 60.6%

    \[\leadsto \color{blue}{\frac{v}{u - t1}} \]
11. Add Preprocessing

Alternative 17: 14.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v t1))

C

double code(double u, double v, double t1) {
	return v / t1;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / t1
end function

Java

public static double code(double u, double v, double t1) {
	return v / t1;
}

Python

def code(u, v, t1):
	return v / t1

Julia

function code(u, v, t1)
	return Float64(v / t1)
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / t1;
end

Wolfram

code[u_, v_, t1_] := N[(v / t1), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 97.6%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   2. distribute-frac-neg 97.6%

      \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

   3. distribute-neg-frac2 97.6%

      \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   4. +-commutative 97.6%

      \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

   5. distribute-neg-in 97.6%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

   6. unsub-neg 97.6%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

3. Simplified 97.6%

   \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing

5. Taylor expanded in t1 around inf 52.7%

   \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{t1 + u} \]

6. Taylor expanded in u around inf 13.8%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))