Rosa's DopplerBench

Percentage Accurate: 72.3% → 97.8%

Time: 7.9s

Alternatives: 10

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: 72.3% 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: 97.8% 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 78.2%

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

   1. times-frac 97.7%

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

3. Simplified 97.7%

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

4. Final simplification 97.7%

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

Alternative 2: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ t_2 := \frac{-t1}{u}\\ t_3 := \frac{v}{t1 + u} \cdot t_2\\ \mathbf{if}\;u \leq -8.8 \cdot 10^{-24}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;u \leq 3 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;t_2 \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 4.4 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1)))
        (t_2 (/ (- t1) u))
        (t_3 (* (/ v (+ t1 u)) t_2)))
   (if (<= u -8.8e-24)
     t_3
     (if (<= u 3e-57)
       t_1
       (if (<= u 1.25e-17) (* t_2 (/ v u)) (if (<= u 4.4e+17) t_1 t_3))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double t_2 = -t1 / u;
	double t_3 = (v / (t1 + u)) * t_2;
	double tmp;
	if (u <= -8.8e-24) {
		tmp = t_3;
	} else if (u <= 3e-57) {
		tmp = t_1;
	} else if (u <= 1.25e-17) {
		tmp = t_2 * (v / u);
	} else if (u <= 4.4e+17) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    t_2 = -t1 / u
    t_3 = (v / (t1 + u)) * t_2
    if (u <= (-8.8d-24)) then
        tmp = t_3
    else if (u <= 3d-57) then
        tmp = t_1
    else if (u <= 1.25d-17) then
        tmp = t_2 * (v / u)
    else if (u <= 4.4d+17) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double t_2 = -t1 / u;
	double t_3 = (v / (t1 + u)) * t_2;
	double tmp;
	if (u <= -8.8e-24) {
		tmp = t_3;
	} else if (u <= 3e-57) {
		tmp = t_1;
	} else if (u <= 1.25e-17) {
		tmp = t_2 * (v / u);
	} else if (u <= 4.4e+17) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	t_2 = -t1 / u
	t_3 = (v / (t1 + u)) * t_2
	tmp = 0
	if u <= -8.8e-24:
		tmp = t_3
	elif u <= 3e-57:
		tmp = t_1
	elif u <= 1.25e-17:
		tmp = t_2 * (v / u)
	elif u <= 4.4e+17:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	t_2 = Float64(Float64(-t1) / u)
	t_3 = Float64(Float64(v / Float64(t1 + u)) * t_2)
	tmp = 0.0
	if (u <= -8.8e-24)
		tmp = t_3;
	elseif (u <= 3e-57)
		tmp = t_1;
	elseif (u <= 1.25e-17)
		tmp = Float64(t_2 * Float64(v / u));
	elseif (u <= 4.4e+17)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	t_2 = -t1 / u;
	t_3 = (v / (t1 + u)) * t_2;
	tmp = 0.0;
	if (u <= -8.8e-24)
		tmp = t_3;
	elseif (u <= 3e-57)
		tmp = t_1;
	elseif (u <= 1.25e-17)
		tmp = t_2 * (v / u);
	elseif (u <= 4.4e+17)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-t1) / u), $MachinePrecision]}, Block[{t$95$3 = N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[u, -8.8e-24], t$95$3, If[LessEqual[u, 3e-57], t$95$1, If[LessEqual[u, 1.25e-17], N[(t$95$2 * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.4e+17], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -8.80000000000000006e-24 or 4.4e17 < u

   1. Initial program 79.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around 0 83.9%

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

      1. associate-*r/ 83.9%

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

      2. mul-1-neg 83.9%

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

   6. Simplified 83.9%

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

   if -8.80000000000000006e-24 < u < 3.00000000000000001e-57 or 1.25e-17 < u < 4.4e17

   1. Initial program 77.2%

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

      1. associate-/r* 83.7%

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

      2. *-commutative 83.7%

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

      3. associate-/l* 97.8%

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

      4. associate-/l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. div-sub 99.9%

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

      9. sub-neg 99.9%

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

      10. *-inverses 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. unsub-neg 86.3%

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

      3. *-commutative 86.3%

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

   6. Simplified 86.3%

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

   if 3.00000000000000001e-57 < u < 1.25e-17

   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 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around 0 78.5%

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

      1. associate-*r/ 78.5%

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

      2. mul-1-neg 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in t1 around 0 78.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 3 \cdot 10^{-57}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 4.4 \cdot 10^{+17}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 3: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.65 \cdot 10^{-107} \lor \neg \left(t1 \leq 3.6 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.65e-107) (not (<= t1 3.6e-50)))
   (/ v (- (* u -2.0) t1))
   (* (/ (- t1) u) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.65e-107) || !(t1 <= 3.6e-50)) {
		tmp = v / ((u * -2.0) - t1);
	} 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 <= (-2.65d-107)) .or. (.not. (t1 <= 3.6d-50))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    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 <= -2.65e-107) || !(t1 <= 3.6e-50)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-t1 / u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.65e-107) or not (t1 <= 3.6e-50):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (-t1 / u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.65e-107) || !(t1 <= 3.6e-50))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.65e-107) || ~((t1 <= 3.6e-50)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (-t1 / u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.65e-107], N[Not[LessEqual[t1, 3.6e-50]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.65e-107 or 3.59999999999999979e-50 < t1

   1. Initial program 76.0%

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

      1. associate-/r* 85.0%

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

      2. *-commutative 85.0%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 95.4%

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

      5. +-commutative 95.4%

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

      6. remove-double-neg 95.4%

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

      7. unsub-neg 95.4%

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

      8. div-sub 95.4%

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

      9. sub-neg 95.4%

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

      10. *-inverses 95.4%

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

      11. metadata-eval 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in t1 around inf 81.0%

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

      1. mul-1-neg 81.0%

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

      2. unsub-neg 81.0%

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

      3. *-commutative 81.0%

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

   6. Simplified 81.0%

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

   if -2.65e-107 < t1 < 3.59999999999999979e-50

   1. Initial program 82.1%

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

      1. times-frac 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in t1 around 0 80.5%

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

      1. associate-*r/ 80.5%

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

      2. mul-1-neg 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in t1 around 0 81.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.65 \cdot 10^{-107} \lor \neg \left(t1 \leq 3.6 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{-108} \lor \neg \left(t1 \leq 3.5 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot \left(-v\right)}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.8e-108) (not (<= t1 3.5e-50)))
   (/ v (- (* u -2.0) t1))
   (/ (* (/ t1 u) (- v)) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.8e-108) || !(t1 <= 3.5e-50)) {
		tmp = v / ((u * -2.0) - t1);
	} 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 <= (-5.8d-108)) .or. (.not. (t1 <= 3.5d-50))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    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 <= -5.8e-108) || !(t1 <= 3.5e-50)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = ((t1 / u) * -v) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.8e-108) or not (t1 <= 3.5e-50):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = ((t1 / u) * -v) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.8e-108) || !(t1 <= 3.5e-50))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(Float64(t1 / u) * Float64(-v)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.8e-108) || ~((t1 <= 3.5e-50)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = ((t1 / u) * -v) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.8e-108], N[Not[LessEqual[t1, 3.5e-50]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] * (-v)), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.8000000000000002e-108 or 3.49999999999999997e-50 < t1

   1. Initial program 76.0%

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

      1. associate-/r* 85.0%

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

      2. *-commutative 85.0%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 95.4%

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

      5. +-commutative 95.4%

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

      6. remove-double-neg 95.4%

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

      7. unsub-neg 95.4%

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

      8. div-sub 95.4%

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

      9. sub-neg 95.4%

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

      10. *-inverses 95.4%

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

      11. metadata-eval 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in t1 around inf 81.0%

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

      1. mul-1-neg 81.0%

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

      2. unsub-neg 81.0%

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

      3. *-commutative 81.0%

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

   6. Simplified 81.0%

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

   if -5.8000000000000002e-108 < t1 < 3.49999999999999997e-50

   1. Initial program 82.1%

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

      1. times-frac 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in t1 around 0 80.5%

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

      1. associate-*r/ 80.5%

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

      2. mul-1-neg 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in t1 around 0 81.0%

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

      1. frac-2neg 81.0%

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

      2. associate-*r/ 81.7%

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

      3. add-sqr-sqrt 36.1%

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

      4. sqrt-unprod 47.1%

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

      5. sqr-neg 47.1%

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

      6. sqrt-unprod 21.4%

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

      7. add-sqr-sqrt 44.3%

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

      8. add-sqr-sqrt 27.0%

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

      9. sqrt-unprod 61.2%

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

      10. sqr-neg 61.2%

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

      11. sqrt-unprod 38.5%

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

      12. add-sqr-sqrt 81.7%

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

   9. Applied egg-rr 81.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{-108} \lor \neg \left(t1 \leq 3.5 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot \left(-v\right)}{u}\\ \end{array} \]

Alternative 5: 67.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{+27} \lor \neg \left(u \leq 3.5 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.3e+27) (not (<= u 3.5e+91)))
   (/ t1 (/ u (/ v u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.3e+27) || !(u <= 3.5e+91)) {
		tmp = t1 / (u / (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+27)) .or. (.not. (u <= 3.5d+91))) then
        tmp = t1 / (u / (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+27) || !(u <= 3.5e+91)) {
		tmp = t1 / (u / (v / u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.3e+27) or not (u <= 3.5e+91):
		tmp = t1 / (u / (v / u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.3e+27) || !(u <= 3.5e+91))
		tmp = Float64(t1 / Float64(u / 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+27) || ~((u <= 3.5e+91)))
		tmp = t1 / (u / (v / u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.3e+27], N[Not[LessEqual[u, 3.5e+91]], $MachinePrecision]], N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.30000000000000008e27 or 3.50000000000000001e91 < u

   1. Initial program 80.6%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around 0 89.1%

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

      1. associate-*r/ 89.1%

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

      2. mul-1-neg 89.1%

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

   6. Simplified 89.1%

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

   7. Taylor expanded in t1 around 0 86.3%

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

      1. associate-*l/ 88.2%

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

      2. associate-/l* 87.3%

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

      3. add-sqr-sqrt 48.2%

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

      4. sqrt-unprod 57.3%

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

      5. sqr-neg 57.3%

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

      6. sqrt-unprod 25.9%

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

      7. add-sqr-sqrt 65.9%

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

   9. Applied egg-rr 65.9%

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

   if -4.30000000000000008e27 < u < 3.50000000000000001e91

   1. Initial program 76.7%

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

      1. times-frac 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in t1 around inf 75.3%

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

      1. associate-*r/ 75.3%

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

      2. neg-mul-1 75.3%

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

   6. Simplified 75.3%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{+27} \lor \neg \left(u \leq 3.5 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 6: 67.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{+27} \lor \neg \left(u \leq 4.2 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.3e+27) (not (<= u 4.2e+93)))
   (/ t1 (/ u (/ v u)))
   (/ v (- (* u -2.0) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.3e+27) || !(u <= 4.2e+93)) {
		tmp = t1 / (u / (v / u));
	} 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) :: tmp
    if ((u <= (-4.3d+27)) .or. (.not. (u <= 4.2d+93))) then
        tmp = t1 / (u / (v / u))
    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 tmp;
	if ((u <= -4.3e+27) || !(u <= 4.2e+93)) {
		tmp = t1 / (u / (v / u));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.3e+27) or not (u <= 4.2e+93):
		tmp = t1 / (u / (v / u))
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.3e+27) || !(u <= 4.2e+93))
		tmp = Float64(t1 / Float64(u / Float64(v / u)));
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.3e+27) || ~((u <= 4.2e+93)))
		tmp = t1 / (u / (v / u));
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.3e+27], N[Not[LessEqual[u, 4.2e+93]], $MachinePrecision]], N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.30000000000000008e27 or 4.1999999999999996e93 < u

   1. Initial program 80.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around 0 89.8%

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

      1. associate-*r/ 89.8%

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

      2. mul-1-neg 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in t1 around 0 87.0%

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

      1. associate-*l/ 89.0%

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

      2. associate-/l* 88.0%

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

      3. add-sqr-sqrt 49.2%

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

      4. sqrt-unprod 57.4%

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

      5. sqr-neg 57.4%

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

      6. sqrt-unprod 25.3%

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

      7. add-sqr-sqrt 66.1%

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

   9. Applied egg-rr 66.1%

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

   if -4.30000000000000008e27 < u < 4.1999999999999996e93

   1. Initial program 77.0%

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

      1. associate-/r* 84.3%

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

      2. *-commutative 84.3%

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

      3. associate-/l* 98.0%

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

      4. associate-/l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. div-sub 99.9%

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

      9. sub-neg 99.9%

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

      10. *-inverses 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 76.7%

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

      1. mul-1-neg 76.7%

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

      2. unsub-neg 76.7%

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

      3. *-commutative 76.7%

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

   6. Simplified 76.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{+27} \lor \neg \left(u \leq 4.2 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 7: 22.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{+199} \lor \neg \left(t1 \leq 2.3 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.5e+199) (not (<= t1 2.3e+57))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e+199) || !(t1 <= 2.3e+57)) {
		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 <= (-1.5d+199)) .or. (.not. (t1 <= 2.3d+57))) 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 <= -1.5e+199) || !(t1 <= 2.3e+57)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.5e+199) or not (t1 <= 2.3e+57):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.5e+199) || !(t1 <= 2.3e+57))
		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 <= -1.5e+199) || ~((t1 <= 2.3e+57)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.5e+199], N[Not[LessEqual[t1, 2.3e+57]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.5e199 or 2.2999999999999999e57 < t1

   1. Initial program 63.1%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 98.6%

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

      3. frac-2neg 98.6%

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

      4. frac-times 77.3%

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

      5. *-un-lft-identity 77.3%

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

      6. remove-double-neg 77.3%

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

      7. distribute-neg-in 77.3%

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

      8. add-sqr-sqrt 24.1%

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

      9. sqrt-unprod 48.9%

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

      10. sqr-neg 48.9%

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

      11. sqrt-unprod 31.1%

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

      12. add-sqr-sqrt 51.3%

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

      13. sub-neg 51.3%

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

   5. Applied egg-rr 51.3%

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

   6. Taylor expanded in t1 around inf 44.4%

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

   if -1.5e199 < t1 < 2.2999999999999999e57

   1. Initial program 84.4%

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

      1. times-frac 96.7%

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

   3. Simplified 96.7%

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

      1. clear-num 96.2%

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

      2. frac-times 94.6%

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

      3. *-un-lft-identity 94.6%

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

      4. frac-2neg 94.6%

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

      5. distribute-neg-in 94.6%

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

      6. add-sqr-sqrt 53.0%

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

      7. sqrt-unprod 76.4%

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

      8. sqr-neg 76.4%

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

      9. sqrt-unprod 26.3%

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

      10. add-sqr-sqrt 57.9%

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

      11. sub-neg 57.9%

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

      12. remove-double-neg 57.9%

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

   5. Applied egg-rr 57.9%

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

      1. associate-/l/ 59.5%

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

   7. Simplified 59.5%

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

   8. Taylor expanded in t1 around 0 60.0%

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

   9. Taylor expanded in u around 0 17.8%

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

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{+199} \lor \neg \left(t1 \leq 2.3 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 8: 61.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return v * (-1.0 / (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 * ((-1.0d0) / (t1 - u))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. frac-times 97.7%

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

   2. associate-*r/ 98.0%

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

   3. div-inv 97.8%

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

   4. clear-num 97.5%

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

   5. associate-*l/ 97.4%

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

   6. *-un-lft-identity 97.4%

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

   7. frac-2neg 97.4%

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

   8. distribute-neg-in 97.4%

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

   9. add-sqr-sqrt 46.1%

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

   10. sqrt-unprod 69.6%

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

   11. sqr-neg 69.6%

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

   12. sqrt-unprod 28.6%

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

   13. add-sqr-sqrt 57.7%

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

   14. sub-neg 57.7%

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

   15. remove-double-neg 57.7%

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

   16. frac-2neg 57.7%

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

   17. metadata-eval 57.7%

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

   18. distribute-neg-in 57.7%

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

   19. add-sqr-sqrt 29.1%

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

   20. sqrt-unprod 72.0%

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

   21. sqr-neg 72.0%

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

   22. sqrt-unprod 50.2%

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

   23. add-sqr-sqrt 95.7%

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

3. Applied egg-rr 95.7%

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

4. Taylor expanded in t1 around inf 63.4%

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

5. Final simplification 63.4%

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

Alternative 9: 55.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= u -3.4e+91) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.4e+91) {
		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+91)) 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+91) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.4e+91:
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.4e+91)
		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 <= -3.4e+91)
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.4e+91], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.4000000000000001e91

   1. Initial program 82.2%

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

      1. times-frac 97.6%

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

   3. Simplified 97.6%

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

      1. clear-num 95.5%

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

      2. frac-times 84.9%

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

      3. *-un-lft-identity 84.9%

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

      4. frac-2neg 84.9%

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

      5. distribute-neg-in 84.9%

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

      6. add-sqr-sqrt 47.9%

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

      7. sqrt-unprod 82.8%

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

      8. sqr-neg 82.8%

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

      9. sqrt-unprod 37.0%

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

      10. add-sqr-sqrt 84.9%

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

      11. sub-neg 84.9%

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

      12. remove-double-neg 84.9%

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

   5. Applied egg-rr 84.9%

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

      1. associate-/l/ 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in t1 around 0 95.5%

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

   9. Taylor expanded in u around 0 48.2%

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

   if -3.4000000000000001e91 < u

   1. Initial program 77.3%

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

      1. times-frac 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in t1 around inf 64.9%

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

      1. associate-*r/ 64.9%

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

      2. neg-mul-1 64.9%

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

   6. Simplified 64.9%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10: 13.9% 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 78.2%

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

   1. times-frac 97.7%

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

3. Simplified 97.7%

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

   1. *-commutative 97.7%

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

   2. clear-num 97.1%

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

   3. frac-2neg 97.1%

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

   4. frac-times 84.8%

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

   5. *-un-lft-identity 84.8%

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

   6. remove-double-neg 84.8%

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

   7. distribute-neg-in 84.8%

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

   8. add-sqr-sqrt 42.4%

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

   9. sqrt-unprod 66.2%

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

   10. sqr-neg 66.2%

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

   11. sqrt-unprod 26.4%

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

   12. add-sqr-sqrt 55.4%

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

   13. sub-neg 55.4%

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

5. Applied egg-rr 55.4%

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

6. Taylor expanded in t1 around inf 17.0%

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

7. Final simplification 17.0%

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

Reproduce

herbie shell --seed 2023319 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))