Rosa's DopplerBench

Percentage Accurate: 72.8% → 98.0%

Time: 9.9s

Alternatives: 11

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: 72.8% 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 80.0%

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

   1. times-frac 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -0.02 \lor \neg \left(u \leq 9.2 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 - u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -0.02) (not (<= u 9.2e-33)))
   (/ (/ t1 (/ (- t1 u) v)) u)
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -0.02) || !(u <= 9.2e-33)) {
		tmp = (t1 / ((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 <= (-0.02d0)) .or. (.not. (u <= 9.2d-33))) then
        tmp = (t1 / ((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 <= -0.02) || !(u <= 9.2e-33)) {
		tmp = (t1 / ((t1 - u) / v)) / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -0.02) or not (u <= 9.2e-33):
		tmp = (t1 / ((t1 - u) / v)) / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -0.02) || !(u <= 9.2e-33))
		tmp = Float64(Float64(t1 / Float64(Float64(t1 - u) / 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 <= -0.02) || ~((u <= 9.2e-33)))
		tmp = (t1 / ((t1 - u) / v)) / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -0.02], N[Not[LessEqual[u, 9.2e-33]], $MachinePrecision]], N[(N[(t1 / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -0.0200000000000000004 or 9.19999999999999942e-33 < u

   1. Initial program 84.1%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around 0 83.6%

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

      1. mul-1-neg 83.6%

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

      2. distribute-neg-frac 83.6%

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

   6. Simplified 83.6%

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

      1. associate-*l/ 85.2%

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

   8. Applied egg-rr 86.1%

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

   if -0.0200000000000000004 < u < 9.19999999999999942e-33

   1. Initial program 74.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around inf 81.7%

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

      1. associate-*r/ 81.7%

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

      2. neg-mul-1 81.7%

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

   6. Simplified 81.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.02 \lor \neg \left(u \leq 9.2 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 - u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 3: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4400000000 \lor \neg \left(t1 \leq 4.2 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4400000000.0) (not (<= t1 4.2e+73)))
   (/ (- v) (+ t1 u))
   (* (/ (- t1) u) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4400000000.0) || !(t1 <= 4.2e+73)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (-t1 / u) * (v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4400000000.0d0)) .or. (.not. (t1 <= 4.2d+73))) then
        tmp = -v / (t1 + u)
    else
        tmp = (-t1 / u) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4400000000.0) || !(t1 <= 4.2e+73)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (-t1 / u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4400000000.0) or not (t1 <= 4.2e+73):
		tmp = -v / (t1 + u)
	else:
		tmp = (-t1 / u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4400000000.0) || !(t1 <= 4.2e+73))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	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 <= -4400000000.0) || ~((t1 <= 4.2e+73)))
		tmp = -v / (t1 + u);
	else
		tmp = (-t1 / u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4400000000.0], N[Not[LessEqual[t1, 4.2e+73]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.4e9 or 4.2000000000000003e73 < t1

   1. Initial program 65.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around inf 89.4%

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

   if -4.4e9 < t1 < 4.2000000000000003e73

   1. Initial program 91.7%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around 0 76.3%

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

      1. mul-1-neg 76.3%

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

      2. distribute-neg-frac 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in t1 around 0 79.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4400000000 \lor \neg \left(t1 \leq 4.2 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 4: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3700000000000 \lor \neg \left(t1 \leq 9.5 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3700000000000.0) (not (<= t1 9.5e+73)))
   (/ (- v) (+ t1 (* u 2.0)))
   (* (/ (- t1) u) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3700000000000.0) || !(t1 <= 9.5e+73)) {
		tmp = -v / (t1 + (u * 2.0));
	} 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 <= (-3700000000000.0d0)) .or. (.not. (t1 <= 9.5d+73))) then
        tmp = -v / (t1 + (u * 2.0d0))
    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 <= -3700000000000.0) || !(t1 <= 9.5e+73)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (-t1 / u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3700000000000.0) or not (t1 <= 9.5e+73):
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = (-t1 / u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3700000000000.0) || !(t1 <= 9.5e+73))
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	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 <= -3700000000000.0) || ~((t1 <= 9.5e+73)))
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = (-t1 / u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3700000000000.0], N[Not[LessEqual[t1, 9.5e+73]], $MachinePrecision]], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.7e12 or 9.4999999999999996e73 < t1

   1. Initial program 65.4%

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

      1. times-frac 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-*l/ 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. remove-double-neg 99.9%

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

      9. unsub-neg 99.9%

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

      10. div-sub 99.9%

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

      11. sub-neg 99.9%

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

      12. distribute-frac-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 97.5%

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

      1. mul-1-neg 97.5%

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

      2. +-commutative 97.5%

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

      3. *-commutative 97.5%

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

      4. distribute-neg-frac 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in t1 around inf 89.9%

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

      1. *-commutative 89.9%

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

   9. Simplified 89.9%

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

   if -3.7e12 < t1 < 9.4999999999999996e73

   1. Initial program 91.7%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around 0 76.3%

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

      1. mul-1-neg 76.3%

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

      2. distribute-neg-frac 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in t1 around 0 79.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3700000000000 \lor \neg \left(t1 \leq 9.5 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 5: 57.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+255} \lor \neg \left(u \leq 1.4 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5e+255) (not (<= u 1.4e+119))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5e+255) || !(u <= 1.4e+119)) {
		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 <= (-5d+255)) .or. (.not. (u <= 1.4d+119))) 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 <= -5e+255) || !(u <= 1.4e+119)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5e+255) or not (u <= 1.4e+119):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5e+255) || !(u <= 1.4e+119))
		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 <= -5e+255) || ~((u <= 1.4e+119)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5e+255], N[Not[LessEqual[u, 1.4e+119]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.0000000000000002e255 or 1.40000000000000007e119 < u

   1. Initial program 87.4%

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

      1. frac-2neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. add-sqr-sqrt 48.4%

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

      6. sqrt-unprod 92.5%

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

      7. sqr-neg 92.5%

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

      8. sqrt-unprod 48.6%

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

      9. add-sqr-sqrt 95.4%

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

      10. sub-neg 95.4%

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

   5. Applied egg-rr 95.4%

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

      1. associate-*r/ 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in t1 around 0 90.9%

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

      1. *-commutative 90.9%

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

   10. Simplified 90.9%

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

   11. Taylor expanded in t1 around inf 40.4%

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

   if -5.0000000000000002e255 < u < 1.40000000000000007e119

   1. Initial program 77.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. neg-mul-1 64.2%

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

   6. Simplified 64.2%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+255} \lor \neg \left(u \leq 1.4 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 6: 57.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+255}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5e+255) (/ v u) (if (<= u 9.5e+101) (/ (- v) t1) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e+255) {
		tmp = v / u;
	} else if (u <= 9.5e+101) {
		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 (u <= (-5d+255)) then
        tmp = v / u
    else if (u <= 9.5d+101) 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 (u <= -5e+255) {
		tmp = v / u;
	} else if (u <= 9.5e+101) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5e+255:
		tmp = v / u
	elif u <= 9.5e+101:
		tmp = -v / t1
	else:
		tmp = -v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5e+255)
		tmp = Float64(v / u);
	elseif (u <= 9.5e+101)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-v) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5e+255)
		tmp = v / u;
	elseif (u <= 9.5e+101)
		tmp = -v / t1;
	else
		tmp = -v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5e+255], N[(v / u), $MachinePrecision], If[LessEqual[u, 9.5e+101], N[((-v) / t1), $MachinePrecision], N[((-v) / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.0000000000000002e255

   1. Initial program 93.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. associate-*l/ 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. add-sqr-sqrt 37.5%

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

      6. sqrt-unprod 94.0%

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

      7. sqr-neg 94.0%

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

      8. sqrt-unprod 62.5%

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

      9. add-sqr-sqrt 100.0%

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

      10. sub-neg 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-*r/ 93.9%

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

   7. Simplified 93.9%

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

   8. Taylor expanded in t1 around 0 93.9%

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

      1. *-commutative 93.9%

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

   10. Simplified 93.9%

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

   11. Taylor expanded in t1 around inf 39.7%

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

   if -5.0000000000000002e255 < u < 9.49999999999999947e101

   1. Initial program 77.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. neg-mul-1 65.1%

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

   6. Simplified 65.1%

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

   if 9.49999999999999947e101 < u

   1. Initial program 86.1%

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

      1. times-frac 99.2%

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

      2. neg-mul-1 99.2%

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

      3. associate-/l* 97.3%

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

      4. associate-*l/ 97.4%

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

      5. neg-mul-1 97.4%

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

      6. distribute-frac-neg 97.4%

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

      7. +-commutative 97.4%

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

      8. remove-double-neg 97.4%

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

      9. unsub-neg 97.4%

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

      10. div-sub 97.4%

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

      11. sub-neg 97.4%

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

      12. distribute-frac-neg 97.4%

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

      13. remove-double-neg 97.4%

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

      14. *-inverses 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in t1 around 0 91.8%

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

      1. associate-*r/ 91.8%

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

      2. neg-mul-1 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in u around 0 38.6%

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

      1. associate-*r/ 38.6%

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

      2. neg-mul-1 38.6%

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

   9. Simplified 38.6%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+255}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]

Alternative 7: 57.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+255}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5e+255)
   (/ 1.0 (/ u v))
   (if (<= u 9.5e+101) (/ (- v) t1) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e+255) {
		tmp = 1.0 / (u / v);
	} else if (u <= 9.5e+101) {
		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 (u <= (-5d+255)) then
        tmp = 1.0d0 / (u / v)
    else if (u <= 9.5d+101) 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 (u <= -5e+255) {
		tmp = 1.0 / (u / v);
	} else if (u <= 9.5e+101) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5e+255:
		tmp = 1.0 / (u / v)
	elif u <= 9.5e+101:
		tmp = -v / t1
	else:
		tmp = -v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5e+255)
		tmp = Float64(1.0 / Float64(u / v));
	elseif (u <= 9.5e+101)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-v) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5e+255)
		tmp = 1.0 / (u / v);
	elseif (u <= 9.5e+101)
		tmp = -v / t1;
	else
		tmp = -v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5e+255], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 9.5e+101], N[((-v) / t1), $MachinePrecision], N[((-v) / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.0000000000000002e255

   1. Initial program 93.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg 99.9%

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

      2. clear-num 99.9%

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

      3. frac-times 94.0%

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

      4. remove-double-neg 94.0%

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

      5. *-commutative 94.0%

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

      6. *-un-lft-identity 94.0%

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

      7. distribute-neg-in 94.0%

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

      8. add-sqr-sqrt 37.5%

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

      9. sqrt-unprod 94.0%

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

      10. sqr-neg 94.0%

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

      11. sqrt-unprod 56.5%

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

      12. add-sqr-sqrt 94.0%

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

      13. sub-neg 94.0%

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

   5. Applied egg-rr 94.0%

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

      1. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t1 around inf 41.4%

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

   9. Taylor expanded in t1 around 0 41.4%

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

   if -5.0000000000000002e255 < u < 9.49999999999999947e101

   1. Initial program 77.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. neg-mul-1 65.1%

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

   6. Simplified 65.1%

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

   if 9.49999999999999947e101 < u

   1. Initial program 86.1%

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

      1. times-frac 99.2%

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

      2. neg-mul-1 99.2%

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

      3. associate-/l* 97.3%

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

      4. associate-*l/ 97.4%

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

      5. neg-mul-1 97.4%

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

      6. distribute-frac-neg 97.4%

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

      7. +-commutative 97.4%

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

      8. remove-double-neg 97.4%

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

      9. unsub-neg 97.4%

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

      10. div-sub 97.4%

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

      11. sub-neg 97.4%

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

      12. distribute-frac-neg 97.4%

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

      13. remove-double-neg 97.4%

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

      14. *-inverses 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in t1 around 0 91.8%

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

      1. associate-*r/ 91.8%

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

      2. neg-mul-1 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in u around 0 38.6%

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

      1. associate-*r/ 38.6%

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

      2. neg-mul-1 38.6%

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

   9. Simplified 38.6%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+255}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]

Alternative 8: 23.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{+72} \lor \neg \left(t1 \leq 5.5 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.4e+72) (not (<= t1 5.5e+110))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.4e+72) || !(t1 <= 5.5e+110)) {
		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 <= (-2.4d+72)) .or. (.not. (t1 <= 5.5d+110))) 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 <= -2.4e+72) || !(t1 <= 5.5e+110)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.4e+72) or not (t1 <= 5.5e+110):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.4e+72) || !(t1 <= 5.5e+110))
		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 <= -2.4e+72) || ~((t1 <= 5.5e+110)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.4e+72], N[Not[LessEqual[t1, 5.5e+110]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.4000000000000001e72 or 5.49999999999999996e110 < t1

   1. Initial program 60.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 97.0%

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

      4. *-un-lft-identity 97.0%

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

      5. add-sqr-sqrt 46.3%

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

      6. sqrt-unprod 62.3%

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

      7. sqr-neg 62.3%

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

      8. sqrt-unprod 24.6%

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

      9. add-sqr-sqrt 45.0%

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

      10. add-sqr-sqrt 21.9%

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

      11. sqrt-unprod 12.2%

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

      12. sqr-neg 12.2%

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

      13. sqrt-unprod 40.1%

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

      14. add-sqr-sqrt 97.0%

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

      15. distribute-neg-in 97.0%

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

      16. add-sqr-sqrt 56.4%

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

      17. sqrt-unprod 61.3%

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

      18. sqr-neg 61.3%

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

      19. sqrt-unprod 23.1%

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

      20. add-sqr-sqrt 47.9%

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

      21. sub-neg 47.9%

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

   5. Applied egg-rr 47.9%

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

   6. Taylor expanded in t1 around inf 39.8%

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

   if -2.4000000000000001e72 < t1 < 5.49999999999999996e110

   1. Initial program 91.9%

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

      1. frac-2neg 97.8%

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

      2. remove-double-neg 97.8%

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

      3. associate-*l/ 98.3%

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

      4. distribute-neg-in 98.3%

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

      5. add-sqr-sqrt 46.0%

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

      6. sqrt-unprod 82.7%

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

      7. sqr-neg 82.7%

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

      8. sqrt-unprod 37.4%

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

      9. add-sqr-sqrt 68.9%

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

      10. sub-neg 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. associate-*r/ 68.2%

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

   7. Simplified 68.2%

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

   8. Taylor expanded in t1 around 0 71.0%

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

      1. *-commutative 71.0%

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

   10. Simplified 71.0%

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

   11. Taylor expanded in t1 around inf 19.1%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{+72} \lor \neg \left(t1 \leq 5.5 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 9: 60.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. times-frac 98.6%

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

3. Simplified 98.6%

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

   1. frac-2neg 98.6%

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

   2. clear-num 97.7%

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

   3. frac-times 83.1%

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

   4. remove-double-neg 83.1%

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

   5. *-commutative 83.1%

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

   6. *-un-lft-identity 83.1%

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

   7. distribute-neg-in 83.1%

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

   8. add-sqr-sqrt 42.2%

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

   9. sqrt-unprod 70.6%

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

   10. sqr-neg 70.6%

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

   11. sqrt-unprod 31.4%

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

   12. add-sqr-sqrt 59.8%

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

   13. sub-neg 59.8%

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

5. Applied egg-rr 59.8%

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

   1. associate-/r* 61.9%

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

7. Simplified 61.9%

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

8. Taylor expanded in t1 around inf 26.3%

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

   1. clear-num 26.1%

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

   2. div-inv 26.1%

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

   3. frac-2neg 26.1%

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

   4. metadata-eval 26.1%

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

   5. distribute-neg-in 26.1%

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

   6. add-sqr-sqrt 13.1%

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

   7. sqrt-unprod 37.7%

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

   8. sqr-neg 37.7%

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

   9. sqrt-unprod 27.9%

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

   10. add-sqr-sqrt 61.2%

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

   11. sub-neg 61.2%

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

10. Applied egg-rr 61.2%

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

    1. *-commutative 61.2%

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

    2. associate-*l/ 61.3%

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

    3. neg-mul-1 61.3%

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

12. Simplified 61.3%

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

13. Final simplification 61.3%

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

Alternative 10: 61.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. times-frac 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in t1 around inf 61.4%

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

5. Final simplification 61.4%

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

Alternative 11: 14.0% 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 80.0%

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

   1. times-frac 98.6%

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

3. Simplified 98.6%

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

   1. clear-num 98.2%

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

   2. frac-2neg 98.2%

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

   3. frac-times 95.9%

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

   4. *-un-lft-identity 95.9%

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

   5. add-sqr-sqrt 42.4%

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

   6. sqrt-unprod 55.0%

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

   7. sqr-neg 55.0%

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

   8. sqrt-unprod 24.0%

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

   9. add-sqr-sqrt 44.2%

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

   10. add-sqr-sqrt 22.3%

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

   11. sqrt-unprod 45.2%

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

   12. sqr-neg 45.2%

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

   13. sqrt-unprod 45.8%

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

   14. add-sqr-sqrt 95.9%

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

   15. distribute-neg-in 95.9%

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

   16. add-sqr-sqrt 49.8%

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

   17. sqrt-unprod 72.7%

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

   18. sqr-neg 72.7%

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

   19. sqrt-unprod 30.2%

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

   20. add-sqr-sqrt 59.1%

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

   21. sub-neg 59.1%

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

5. Applied egg-rr 59.1%

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

6. Taylor expanded in t1 around inf 16.8%

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

7. Final simplification 16.8%

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

Reproduce

herbie shell --seed 2023307 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))