Rosa's DopplerBench

Percentage Accurate: 73.5% → 98.0%

Time: 7.9s

Alternatives: 12

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: 73.5% 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: 98.0% accurate, 1.0× speedup

Math

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

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(Float64(-t1) / 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 67.0%

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

   1. times-frac 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -0.0135:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 0.0036 \lor \neg \left(u \leq 1.1 \cdot 10^{+70}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- t1) u) (/ v u))))
   (if (<= u -0.0135)
     t_1
     (if (<= u 2.3e-80)
       (/ (- v) t1)
       (if (or (<= u 0.0036) (not (<= u 1.1e+70)))
         t_1
         (/ v (- (* u -2.0) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -0.0135) {
		tmp = t_1;
	} else if (u <= 2.3e-80) {
		tmp = -v / t1;
	} else if ((u <= 0.0036) || !(u <= 1.1e+70)) {
		tmp = t_1;
	} else {
		tmp = v / ((u * -2.0) - 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 = (-t1 / u) * (v / u)
    if (u <= (-0.0135d0)) then
        tmp = t_1
    else if (u <= 2.3d-80) then
        tmp = -v / t1
    else if ((u <= 0.0036d0) .or. (.not. (u <= 1.1d+70))) then
        tmp = t_1
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -0.0135) {
		tmp = t_1;
	} else if (u <= 2.3e-80) {
		tmp = -v / t1;
	} else if ((u <= 0.0036) || !(u <= 1.1e+70)) {
		tmp = t_1;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 / u) * (v / u)
	tmp = 0
	if u <= -0.0135:
		tmp = t_1
	elif u <= 2.3e-80:
		tmp = -v / t1
	elif (u <= 0.0036) or not (u <= 1.1e+70):
		tmp = t_1
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) / u) * Float64(v / u))
	tmp = 0.0
	if (u <= -0.0135)
		tmp = t_1;
	elseif (u <= 2.3e-80)
		tmp = Float64(Float64(-v) / t1);
	elseif ((u <= 0.0036) || !(u <= 1.1e+70))
		tmp = t_1;
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 / u) * (v / u);
	tmp = 0.0;
	if (u <= -0.0135)
		tmp = t_1;
	elseif (u <= 2.3e-80)
		tmp = -v / t1;
	elseif ((u <= 0.0036) || ~((u <= 1.1e+70)))
		tmp = t_1;
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -0.0135], t$95$1, If[LessEqual[u, 2.3e-80], N[((-v) / t1), $MachinePrecision], If[Or[LessEqual[u, 0.0036], N[Not[LessEqual[u, 1.1e+70]], $MachinePrecision]], t$95$1, N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -0.0134999999999999998 or 2.2999999999999998e-80 < u < 0.0035999999999999999 or 1.1e70 < u

   1. Initial program 72.3%

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

      1. times-frac 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around 0 84.0%

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

      1. associate-*r/ 84.0%

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

      2. mul-1-neg 84.0%

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

   7. Simplified 84.0%

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

   8. Taylor expanded in t1 around 0 81.9%

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

   if -0.0134999999999999998 < u < 2.2999999999999998e-80

   1. Initial program 60.0%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

   7. Simplified 86.1%

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

   if 0.0035999999999999999 < u < 1.1e70

   1. Initial program 70.9%

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

      1. associate-/r* 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-/l* 99.7%

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

      4. associate-/l/ 99.7%

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

      5. +-commutative 99.7%

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

      6. remove-double-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. div-sub 99.7%

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

      9. sub-neg 99.7%

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

      10. *-inverses 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t1 around inf 72.0%

      \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]
   6. Step-by-step derivation

      1. mul-1-neg 72.0%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 72.0%

         \[\leadsto \frac{v}{\color{blue}{-2 \cdot u - t1}} \]

      3. *-commutative 72.0%

         \[\leadsto \frac{v}{\color{blue}{u \cdot -2} - t1} \]

   7. Simplified 72.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.0135:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 0.0036 \lor \neg \left(u \leq 1.1 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -0.15:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 0.018:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- t1) u) (/ v u))))
   (if (<= u -0.15)
     t_1
     (if (<= u 2.3e-80)
       (/ (- v) t1)
       (if (<= u 0.018)
         (* v (/ t1 (* u (- t1 u))))
         (if (<= u 9.5e+69) (/ v (- (* u -2.0) t1)) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -0.15) {
		tmp = t_1;
	} else if (u <= 2.3e-80) {
		tmp = -v / t1;
	} else if (u <= 0.018) {
		tmp = v * (t1 / (u * (t1 - u)));
	} else if (u <= 9.5e+69) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t_1;
	}
	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 = (-t1 / u) * (v / u)
    if (u <= (-0.15d0)) then
        tmp = t_1
    else if (u <= 2.3d-80) then
        tmp = -v / t1
    else if (u <= 0.018d0) then
        tmp = v * (t1 / (u * (t1 - u)))
    else if (u <= 9.5d+69) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -0.15) {
		tmp = t_1;
	} else if (u <= 2.3e-80) {
		tmp = -v / t1;
	} else if (u <= 0.018) {
		tmp = v * (t1 / (u * (t1 - u)));
	} else if (u <= 9.5e+69) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 / u) * (v / u)
	tmp = 0
	if u <= -0.15:
		tmp = t_1
	elif u <= 2.3e-80:
		tmp = -v / t1
	elif u <= 0.018:
		tmp = v * (t1 / (u * (t1 - u)))
	elif u <= 9.5e+69:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) / u) * Float64(v / u))
	tmp = 0.0
	if (u <= -0.15)
		tmp = t_1;
	elseif (u <= 2.3e-80)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 0.018)
		tmp = Float64(v * Float64(t1 / Float64(u * Float64(t1 - u))));
	elseif (u <= 9.5e+69)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 / u) * (v / u);
	tmp = 0.0;
	if (u <= -0.15)
		tmp = t_1;
	elseif (u <= 2.3e-80)
		tmp = -v / t1;
	elseif (u <= 0.018)
		tmp = v * (t1 / (u * (t1 - u)));
	elseif (u <= 9.5e+69)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -0.15], t$95$1, If[LessEqual[u, 2.3e-80], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 0.018], N[(v * N[(t1 / N[(u * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 9.5e+69], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -0.149999999999999994 or 9.4999999999999995e69 < u

   1. Initial program 69.5%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 85.0%

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

      1. associate-*r/ 85.0%

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

      2. mul-1-neg 85.0%

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

   7. Simplified 85.0%

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

   8. Taylor expanded in t1 around 0 83.4%

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

   if -0.149999999999999994 < u < 2.2999999999999998e-80

   1. Initial program 60.0%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

   7. Simplified 86.1%

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

   if 2.2999999999999998e-80 < u < 0.0179999999999999986

   1. Initial program 89.9%

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

      1. times-frac 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t1 around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. mul-1-neg 77.7%

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

   7. Simplified 77.7%

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

      1. associate-*r/ 72.7%

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

      2. frac-2neg 72.7%

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

      3. div-inv 72.7%

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

      4. div-inv 72.7%

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

      5. add-sqr-sqrt 22.1%

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

      6. sqrt-unprod 29.9%

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

      7. sqr-neg 29.9%

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

      8. sqrt-unprod 18.2%

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

      9. add-sqr-sqrt 30.4%

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

      10. distribute-lft-neg-out 30.4%

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

      11. distribute-frac-neg 30.4%

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

      12. *-commutative 30.4%

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

      13. add-sqr-sqrt 12.2%

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

      14. sqrt-unprod 41.4%

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

      15. sqr-neg 41.4%

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

      16. sqrt-unprod 50.4%

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

      17. add-sqr-sqrt 72.7%

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

      18. distribute-neg-in 72.7%

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

   9. Applied egg-rr 73.2%

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

       1. associate-/l* 73.2%

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

   11. Simplified 73.2%

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

       1. clear-num 73.1%

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

       2. associate-/r/ 73.2%

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

       3. clear-num 73.3%

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

       4. associate-/l/ 73.3%

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

   13. Applied egg-rr 73.3%

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

   if 0.0179999999999999986 < u < 9.4999999999999995e69

   1. Initial program 70.9%

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

      1. associate-/r* 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-/l* 99.7%

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

      4. associate-/l/ 99.7%

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

      5. +-commutative 99.7%

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

      6. remove-double-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. div-sub 99.7%

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

      9. sub-neg 99.7%

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

      10. *-inverses 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t1 around inf 72.0%

      \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]
   6. Step-by-step derivation

      1. mul-1-neg 72.0%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 72.0%

         \[\leadsto \frac{v}{\color{blue}{-2 \cdot u - t1}} \]

      3. *-commutative 72.0%

         \[\leadsto \frac{v}{\color{blue}{u \cdot -2} - t1} \]

   7. Simplified 72.0%

      \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.15:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 0.018:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -0.0021:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 110:\\ \;\;\;\;\frac{v}{u \cdot \frac{t1 - u}{t1}}\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{+70}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- t1) u) (/ v u))))
   (if (<= u -0.0021)
     t_1
     (if (<= u 2.3e-80)
       (/ (- v) t1)
       (if (<= u 110.0)
         (/ v (* u (/ (- t1 u) t1)))
         (if (<= u 1.05e+70) (/ v (- (* u -2.0) t1)) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -0.0021) {
		tmp = t_1;
	} else if (u <= 2.3e-80) {
		tmp = -v / t1;
	} else if (u <= 110.0) {
		tmp = v / (u * ((t1 - u) / t1));
	} else if (u <= 1.05e+70) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t_1;
	}
	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 = (-t1 / u) * (v / u)
    if (u <= (-0.0021d0)) then
        tmp = t_1
    else if (u <= 2.3d-80) then
        tmp = -v / t1
    else if (u <= 110.0d0) then
        tmp = v / (u * ((t1 - u) / t1))
    else if (u <= 1.05d+70) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -0.0021) {
		tmp = t_1;
	} else if (u <= 2.3e-80) {
		tmp = -v / t1;
	} else if (u <= 110.0) {
		tmp = v / (u * ((t1 - u) / t1));
	} else if (u <= 1.05e+70) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 / u) * (v / u)
	tmp = 0
	if u <= -0.0021:
		tmp = t_1
	elif u <= 2.3e-80:
		tmp = -v / t1
	elif u <= 110.0:
		tmp = v / (u * ((t1 - u) / t1))
	elif u <= 1.05e+70:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) / u) * Float64(v / u))
	tmp = 0.0
	if (u <= -0.0021)
		tmp = t_1;
	elseif (u <= 2.3e-80)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 110.0)
		tmp = Float64(v / Float64(u * Float64(Float64(t1 - u) / t1)));
	elseif (u <= 1.05e+70)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 / u) * (v / u);
	tmp = 0.0;
	if (u <= -0.0021)
		tmp = t_1;
	elseif (u <= 2.3e-80)
		tmp = -v / t1;
	elseif (u <= 110.0)
		tmp = v / (u * ((t1 - u) / t1));
	elseif (u <= 1.05e+70)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -0.0021], t$95$1, If[LessEqual[u, 2.3e-80], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 110.0], N[(v / N[(u * N[(N[(t1 - u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.05e+70], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -0.00209999999999999987 or 1.05000000000000004e70 < u

   1. Initial program 69.5%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 85.0%

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

      1. associate-*r/ 85.0%

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

      2. mul-1-neg 85.0%

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

   7. Simplified 85.0%

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

   8. Taylor expanded in t1 around 0 83.4%

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

   if -0.00209999999999999987 < u < 2.2999999999999998e-80

   1. Initial program 60.0%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

   7. Simplified 86.1%

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

   if 2.2999999999999998e-80 < u < 110

   1. Initial program 89.9%

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

      1. times-frac 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t1 around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. mul-1-neg 77.7%

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

   7. Simplified 77.7%

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

      1. associate-*r/ 72.7%

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

      2. frac-2neg 72.7%

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

      3. div-inv 72.7%

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

      4. div-inv 72.7%

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

      5. add-sqr-sqrt 22.1%

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

      6. sqrt-unprod 29.9%

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

      7. sqr-neg 29.9%

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

      8. sqrt-unprod 18.2%

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

      9. add-sqr-sqrt 30.4%

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

      10. distribute-lft-neg-out 30.4%

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

      11. distribute-frac-neg 30.4%

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

      12. *-commutative 30.4%

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

      13. add-sqr-sqrt 12.2%

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

      14. sqrt-unprod 41.4%

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

      15. sqr-neg 41.4%

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

      16. sqrt-unprod 50.4%

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

      17. add-sqr-sqrt 72.7%

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

      18. distribute-neg-in 72.7%

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

   9. Applied egg-rr 73.2%

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

       1. associate-/l* 73.2%

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

   11. Simplified 73.2%

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

       1. associate-/r/ 73.5%

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

   13. Applied egg-rr 73.5%

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

   if 110 < u < 1.05000000000000004e70

   1. Initial program 70.9%

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

      1. associate-/r* 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-/l* 99.7%

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

      4. associate-/l/ 99.7%

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

      5. +-commutative 99.7%

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

      6. remove-double-neg 99.7%

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

      7. unsub-neg 99.7%

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

      8. div-sub 99.7%

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

      9. sub-neg 99.7%

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

      10. *-inverses 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t1 around inf 72.0%

      \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]
   6. Step-by-step derivation

      1. mul-1-neg 72.0%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 72.0%

         \[\leadsto \frac{v}{\color{blue}{-2 \cdot u - t1}} \]

      3. *-commutative 72.0%

         \[\leadsto \frac{v}{\color{blue}{u \cdot -2} - t1} \]

   7. Simplified 72.0%

      \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.0021:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 110:\\ \;\;\;\;\frac{v}{u \cdot \frac{t1 - u}{t1}}\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{+70}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.36 \cdot 10^{-27} \lor \neg \left(u \leq 2.3 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{\frac{v}{\frac{t1 - u}{t1}}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.36e-27) (not (<= u 2.3e-80)))
   (/ (/ v (/ (- t1 u) t1)) (+ t1 u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.36e-27) || !(u <= 2.3e-80)) {
		tmp = (v / ((t1 - u) / t1)) / (t1 + 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 <= (-1.36d-27)) .or. (.not. (u <= 2.3d-80))) then
        tmp = (v / ((t1 - u) / t1)) / (t1 + 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 <= -1.36e-27) || !(u <= 2.3e-80)) {
		tmp = (v / ((t1 - u) / t1)) / (t1 + u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.36e-27) or not (u <= 2.3e-80):
		tmp = (v / ((t1 - u) / t1)) / (t1 + u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.36e-27) || !(u <= 2.3e-80))
		tmp = Float64(Float64(v / Float64(Float64(t1 - u) / t1)) / Float64(t1 + u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.36e-27) || ~((u <= 2.3e-80)))
		tmp = (v / ((t1 - u) / t1)) / (t1 + u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.36e-27], N[Not[LessEqual[u, 2.3e-80]], $MachinePrecision]], N[(N[(v / N[(N[(t1 - u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.36e-27 or 2.2999999999999998e-80 < u

   1. Initial program 73.2%

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

      1. times-frac 98.7%

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

   3. Simplified 98.7%

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

      1. associate-*r/ 98.6%

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

      2. clear-num 98.5%

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

      3. associate-*l/ 98.5%

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

      4. *-un-lft-identity 98.5%

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

      5. frac-2neg 98.5%

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

      6. distribute-neg-in 98.5%

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

      7. add-sqr-sqrt 48.5%

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

      8. sqrt-unprod 83.2%

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

      9. sqr-neg 83.2%

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

      10. sqrt-unprod 41.3%

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

      11. add-sqr-sqrt 80.6%

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

      12. sub-neg 80.6%

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

      13. remove-double-neg 80.6%

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

   6. Applied egg-rr 80.6%

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

   if -1.36e-27 < u < 2.2999999999999998e-80

   1. Initial program 57.6%

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

      1. times-frac 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in t1 around inf 87.1%

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

      1. associate-*r/ 87.1%

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

      2. neg-mul-1 87.1%

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

   7. Simplified 87.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.36 \cdot 10^{-27} \lor \neg \left(u \leq 2.3 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{\frac{v}{\frac{t1 - u}{t1}}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -0.025 \lor \neg \left(u \leq 2.3 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -0.025) (not (<= u 2.3e-80)))
   (* (/ v (+ t1 u)) (/ (- t1) u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -0.025) || !(u <= 2.3e-80)) {
		tmp = (v / (t1 + u)) * (-t1 / 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 <= (-0.025d0)) .or. (.not. (u <= 2.3d-80))) then
        tmp = (v / (t1 + u)) * (-t1 / 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 <= -0.025) || !(u <= 2.3e-80)) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -0.025) or not (u <= 2.3e-80):
		tmp = (v / (t1 + u)) * (-t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -0.025) || !(u <= 2.3e-80))
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -0.025) || ~((u <= 2.3e-80)))
		tmp = (v / (t1 + u)) * (-t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -0.025], N[Not[LessEqual[u, 2.3e-80]], $MachinePrecision]], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -0.025000000000000001 or 2.2999999999999998e-80 < u

   1. Initial program 72.2%

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

      1. times-frac 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around 0 80.6%

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

      1. associate-*r/ 80.6%

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

      2. mul-1-neg 80.6%

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

   7. Simplified 80.6%

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

   if -0.025000000000000001 < u < 2.2999999999999998e-80

   1. Initial program 60.0%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.025 \lor \neg \left(u \leq 2.3 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 23.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.9 \cdot 10^{+126} \lor \neg \left(t1 \leq 2.3 \cdot 10^{+114}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.9e+126) (not (<= t1 2.3e+114))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.9e+126) || !(t1 <= 2.3e+114)) {
		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 <= (-4.9d+126)) .or. (.not. (t1 <= 2.3d+114))) 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 <= -4.9e+126) || !(t1 <= 2.3e+114)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.9e+126) or not (t1 <= 2.3e+114):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.9e+126) || !(t1 <= 2.3e+114))
		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 <= -4.9e+126) || ~((t1 <= 2.3e+114)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.9e+126], N[Not[LessEqual[t1, 2.3e+114]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.90000000000000001e126 or 2.3e114 < t1

   1. Initial program 41.3%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 91.8%

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

   6. Taylor expanded in u around inf 36.9%

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

   if -4.90000000000000001e126 < t1 < 2.3e114

   1. Initial program 78.7%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 69.0%

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

      1. associate-*r/ 69.0%

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

      2. mul-1-neg 69.0%

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

   7. Simplified 69.0%

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

      1. associate-*r/ 68.4%

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

      2. frac-2neg 68.4%

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

      3. div-inv 68.4%

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

      4. div-inv 68.4%

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

      5. add-sqr-sqrt 33.4%

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

      6. sqrt-unprod 45.7%

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

      7. sqr-neg 45.7%

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

      8. sqrt-unprod 16.6%

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

      9. add-sqr-sqrt 34.1%

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

      10. distribute-lft-neg-out 34.1%

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

      11. distribute-frac-neg 34.1%

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

      12. *-commutative 34.1%

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

      13. add-sqr-sqrt 17.5%

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

      14. sqrt-unprod 44.5%

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

      15. sqr-neg 44.5%

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

      16. sqrt-unprod 34.8%

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

      17. add-sqr-sqrt 68.4%

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

      18. distribute-neg-in 68.4%

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

   9. Applied egg-rr 67.8%

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

   10. Taylor expanded in t1 around inf 16.6%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.9 \cdot 10^{+126} \lor \neg \left(t1 \leq 2.3 \cdot 10^{+114}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7 \cdot 10^{+148}:\\ \;\;\;\;v \cdot \frac{-0.5}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7e+148) (* v (/ -0.5 u)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7e+148) {
		tmp = v * (-0.5 / 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 <= (-7d+148)) then
        tmp = v * ((-0.5d0) / 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 <= -7e+148) {
		tmp = v * (-0.5 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7e+148:
		tmp = v * (-0.5 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7e+148)
		tmp = Float64(v * Float64(-0.5 / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -7e+148)
		tmp = v * (-0.5 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7e+148], N[(v * N[(-0.5 / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.9999999999999998e148

   1. Initial program 62.8%

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

      1. associate-/r* 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-/l* 97.3%

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

      4. associate-/l/ 77.9%

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

      5. +-commutative 77.9%

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

      6. remove-double-neg 77.9%

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

      7. unsub-neg 77.9%

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

      8. div-sub 77.9%

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

      9. sub-neg 77.9%

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

      10. *-inverses 77.9%

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

      11. metadata-eval 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in t1 around inf 47.2%

      \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]
   6. Step-by-step derivation

      1. mul-1-neg 47.2%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 47.2%

         \[\leadsto \frac{v}{\color{blue}{-2 \cdot u - t1}} \]

      3. *-commutative 47.2%

         \[\leadsto \frac{v}{\color{blue}{u \cdot -2} - t1} \]

   7. Simplified 47.2%

      \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]
   8. Step-by-step derivation

      1. div-inv 47.2%

         \[\leadsto \color{blue}{v \cdot \frac{1}{u \cdot -2 - t1}} \]

   9. Applied egg-rr 47.2%

      \[\leadsto \color{blue}{v \cdot \frac{1}{u \cdot -2 - t1}} \]

   10. Taylor expanded in u around inf 44.7%

       \[\leadsto v \cdot \color{blue}{\frac{-0.5}{u}} \]

   if -6.9999999999999998e148 < u

   1. Initial program 67.8%

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

      1. times-frac 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in t1 around inf 60.0%

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

      1. associate-*r/ 60.0%

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

      2. neg-mul-1 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7 \cdot 10^{+148}:\\ \;\;\;\;v \cdot \frac{-0.5}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u \cdot -2 - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- (* u -2.0) t1)))

C

double code(double u, double v, double t1) {
	return v / ((u * -2.0) - 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 * (-2.0d0)) - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Python

def code(u, v, t1):
	return v / ((u * -2.0) - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(u * -2.0) - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((u * -2.0) - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.0%

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

   1. associate-/r* 78.4%

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

   2. *-commutative 78.4%

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

   3. associate-/l* 98.3%

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

   4. associate-/l/ 94.4%

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

   5. +-commutative 94.4%

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

   6. remove-double-neg 94.4%

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

   7. unsub-neg 94.4%

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

   8. div-sub 94.4%

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

   9. sub-neg 94.4%

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

   10. *-inverses 94.4%

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

   11. metadata-eval 94.4%

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

3. Simplified 94.4%

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

5. Taylor expanded in t1 around inf 61.1%

   \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]
6. Step-by-step derivation

   1. mul-1-neg 61.1%

      \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

   2. unsub-neg 61.1%

      \[\leadsto \frac{v}{\color{blue}{-2 \cdot u - t1}} \]

   3. *-commutative 61.1%

      \[\leadsto \frac{v}{\color{blue}{u \cdot -2} - t1} \]

7. Simplified 61.1%

   \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]

8. Final simplification 61.1%

   \[\leadsto \frac{v}{u \cdot -2 - t1} \]
9. Add Preprocessing

Alternative 10: 56.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{+151}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.3e+151) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.3e+151) {
		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 <= (-4.3d+151)) 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 <= -4.3e+151) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.3e+151:
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.3e+151)
		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 <= -4.3e+151)
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.3e+151], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.29999999999999983e151

   1. Initial program 61.8%

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

      1. times-frac 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in t1 around 0 94.8%

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

      1. associate-*r/ 94.8%

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

      2. mul-1-neg 94.8%

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

   7. Simplified 94.8%

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

      1. associate-*r/ 94.8%

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

      2. frac-2neg 94.8%

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

      3. div-inv 94.8%

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

      4. div-inv 94.8%

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

      5. add-sqr-sqrt 48.2%

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

      6. sqrt-unprod 66.1%

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

      7. sqr-neg 66.1%

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

      8. sqrt-unprod 30.8%

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

      9. add-sqr-sqrt 61.6%

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

      10. distribute-lft-neg-out 61.6%

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

      11. distribute-frac-neg 61.6%

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

      12. *-commutative 61.6%

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

      13. add-sqr-sqrt 30.8%

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

      14. sqrt-unprod 66.5%

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

      15. sqr-neg 66.5%

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

      16. sqrt-unprod 46.4%

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

      17. add-sqr-sqrt 94.8%

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

      18. distribute-neg-in 94.8%

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

   9. Applied egg-rr 94.7%

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

   10. Taylor expanded in t1 around inf 45.5%

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

   if -4.29999999999999983e151 < u

   1. Initial program 67.9%

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

      1. times-frac 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in t1 around inf 59.7%

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

      1. associate-*r/ 59.7%

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

      2. neg-mul-1 59.7%

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

   7. Simplified 59.7%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{+151}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.5e+148) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.5e+148) {
		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 <= (-3.5d+148)) 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 <= -3.5e+148) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.5e+148:
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.5e+148)
		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 <= -3.5e+148)
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.5e+148], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.4999999999999999e148

   1. Initial program 62.8%

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

      1. times-frac 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in t1 around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. mul-1-neg 94.9%

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

   7. Simplified 94.9%

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

   8. Taylor expanded in t1 around inf 44.6%

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

      1. associate-*r/ 44.6%

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

      2. neg-mul-1 44.6%

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

   10. Simplified 44.6%

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

   if -3.4999999999999999e148 < u

   1. Initial program 67.8%

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

      1. times-frac 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in t1 around inf 60.0%

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

      1. associate-*r/ 60.0%

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

      2. neg-mul-1 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 14.5% 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 67.0%

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

   1. times-frac 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in t1 around inf 55.1%

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

6. Taylor expanded in u around inf 14.9%

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

7. Final simplification 14.9%

   \[\leadsto \frac{v}{t1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))