Rosa's DopplerBench

Percentage Accurate: 72.9% → 98.1%

Time: 8.9s

Alternatives: 14

Speedup: 1.1×

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: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

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 74.2%

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

   1. times-frac 97.5%

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

3. Simplified 97.5%

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

4. Final simplification 97.5%

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

Alternative 2: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+48} \lor \neg \left(u \leq 1.5 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.9e+48) (not (<= u 1.5e+165)))
   (/ (/ t1 u) (/ u v))
   (/ (- v) (+ t1 (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.9e+48) || !(u <= 1.5e+165)) {
		tmp = (t1 / u) / (u / v);
	} else {
		tmp = -v / (t1 + (u * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.9d+48)) .or. (.not. (u <= 1.5d+165))) then
        tmp = (t1 / u) / (u / v)
    else
        tmp = -v / (t1 + (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.9e+48) || !(u <= 1.5e+165)) {
		tmp = (t1 / u) / (u / v);
	} else {
		tmp = -v / (t1 + (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.9e+48) or not (u <= 1.5e+165):
		tmp = (t1 / u) / (u / v)
	else:
		tmp = -v / (t1 + (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.9e+48) || !(u <= 1.5e+165))
		tmp = Float64(Float64(t1 / u) / Float64(u / v));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.9e+48) || ~((u <= 1.5e+165)))
		tmp = (t1 / u) / (u / v);
	else
		tmp = -v / (t1 + (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.9e+48], N[Not[LessEqual[u, 1.5e+165]], $MachinePrecision]], N[(N[(t1 / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.9e48 or 1.49999999999999995e165 < u

   1. Initial program 78.6%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in t1 around 0 81.9%

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

   5. Taylor expanded in t1 around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. neg-mul-1 81.9%

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

   7. Simplified 81.9%

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

      1. clear-num 81.8%

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

      2. un-div-inv 81.8%

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

      3. add-sqr-sqrt 40.9%

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

      4. sqrt-unprod 59.2%

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

      5. sqr-neg 59.2%

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

      6. sqrt-unprod 32.9%

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

      7. add-sqr-sqrt 64.5%

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

   9. Applied egg-rr 64.5%

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

   if -1.9e48 < u < 1.49999999999999995e165

   1. Initial program 72.2%

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

      1. *-commutative 72.2%

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

      2. times-frac 98.0%

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

      3. neg-mul-1 98.0%

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

      4. associate-/l* 97.6%

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

      5. associate-*r/ 97.6%

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

      6. associate-/l* 97.6%

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

      7. associate-/l/ 97.6%

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

      8. neg-mul-1 97.6%

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

      9. *-lft-identity 97.6%

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

      10. metadata-eval 97.6%

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

      11. times-frac 97.6%

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

      12. neg-mul-1 97.6%

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

      13. remove-double-neg 97.6%

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

      14. neg-mul-1 97.6%

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

      15. sub0-neg 97.6%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 97.6%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 97.6%

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

      18. div-sub 97.6%

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

      19. distribute-frac-neg 97.6%

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

      20. *-inverses 97.6%

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

      21. metadata-eval 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in v around 0 96.6%

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

      1. mul-1-neg 96.6%

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

      2. +-commutative 96.6%

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

   6. Simplified 96.6%

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

   7. Taylor expanded in t1 around inf 68.7%

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

      1. *-commutative 68.7%

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

   9. Simplified 68.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+48} \lor \neg \left(u \leq 1.5 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]

Alternative 3: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{+14} \lor \neg \left(t1 \leq 0.000125\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.1e+14) (not (<= t1 0.000125)))
   (/ (- v) (+ t1 (* u 2.0)))
   (* t1 (/ (- v) (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.1e+14) || !(t1 <= 0.000125)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t1 * (-v / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.1d+14)) .or. (.not. (t1 <= 0.000125d0))) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = t1 * (-v / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.1e+14) || !(t1 <= 0.000125)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t1 * (-v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.1e+14) or not (t1 <= 0.000125):
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = t1 * (-v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.1e+14) || !(t1 <= 0.000125))
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.1e+14) || ~((t1 <= 0.000125)))
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = t1 * (-v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.1e+14], N[Not[LessEqual[t1, 0.000125]], $MachinePrecision]], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.1e14 or 1.25e-4 < t1

   1. Initial program 64.9%

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

      1. *-commutative 64.9%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. +-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in t1 around inf 87.7%

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

      1. *-commutative 87.7%

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

   9. Simplified 87.7%

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

   if -1.1e14 < t1 < 1.25e-4

   1. Initial program 83.3%

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

      1. associate-/l* 84.2%

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

      2. neg-mul-1 84.2%

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

      3. *-commutative 84.2%

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

      4. associate-*r/ 83.7%

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

      5. associate-/l* 83.7%

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

      6. neg-mul-1 83.7%

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

      7. associate-/r* 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in t1 around 0 71.6%

      \[\leadsto t1 \cdot \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

      3. unpow2 71.6%

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

   6. Simplified 71.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{+14} \lor \neg \left(t1 \leq 0.000125\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 4: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{+14} \lor \neg \left(t1 \leq 0.000205\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.05e+14) (not (<= t1 0.000205)))
   (/ (- v) (+ t1 (* u 2.0)))
   (* (/ v u) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.05e+14) || !(t1 <= 0.000205)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (v / u) * (-t1 / 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 <= (-1.05d+14)) .or. (.not. (t1 <= 0.000205d0))) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = (v / u) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.05e+14) || !(t1 <= 0.000205)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.05e+14) or not (t1 <= 0.000205):
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.05e+14) || !(t1 <= 0.000205))
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.05e+14) || ~((t1 <= 0.000205)))
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.05e+14], N[Not[LessEqual[t1, 0.000205]], $MachinePrecision]], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.05e14 or 2.05e-4 < t1

   1. Initial program 64.9%

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

      1. *-commutative 64.9%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. +-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in t1 around inf 87.7%

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

      1. *-commutative 87.7%

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

   9. Simplified 87.7%

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

   if -1.05e14 < t1 < 2.05e-4

   1. Initial program 83.3%

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

      1. times-frac 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in t1 around 0 73.8%

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

   5. Taylor expanded in t1 around 0 76.4%

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

      1. associate-*r/ 76.4%

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

      2. neg-mul-1 76.4%

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

   7. Simplified 76.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{+14} \lor \neg \left(t1 \leq 0.000205\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{+14} \lor \neg \left(t1 \leq 13000\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1e+14) (not (<= t1 13000.0)))
   (/ (- v) (+ t1 (* u 2.0)))
   (/ (* t1 (/ v u)) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e+14) || !(t1 <= 13000.0)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1d+14)) .or. (.not. (t1 <= 13000.0d0))) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = (t1 * (v / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e+14) || !(t1 <= 13000.0)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1e+14) or not (t1 <= 13000.0):
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = (t1 * (v / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1e+14) || !(t1 <= 13000.0))
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1e+14) || ~((t1 <= 13000.0)))
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = (t1 * (v / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1e+14], N[Not[LessEqual[t1, 13000.0]], $MachinePrecision]], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1e14 or 13000 < t1

   1. Initial program 64.9%

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

      1. *-commutative 64.9%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. +-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in t1 around inf 87.7%

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

      1. *-commutative 87.7%

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

   9. Simplified 87.7%

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

   if -1e14 < t1 < 13000

   1. Initial program 83.3%

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

      1. times-frac 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in t1 around 0 73.8%

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

   5. Taylor expanded in t1 around 0 76.4%

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

      1. associate-*r/ 76.4%

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

      2. neg-mul-1 76.4%

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

   7. Simplified 76.4%

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

      1. associate-*l/ 79.0%

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

      2. frac-2neg 79.0%

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

      3. add-sqr-sqrt 37.4%

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

      4. sqrt-unprod 49.4%

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

      5. sqr-neg 49.4%

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

      6. sqrt-unprod 20.4%

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

      7. add-sqr-sqrt 37.3%

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

      8. distribute-lft-neg-out 37.3%

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

      9. add-sqr-sqrt 16.8%

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

      10. sqrt-unprod 44.1%

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

      11. sqr-neg 44.1%

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

      12. sqrt-unprod 41.4%

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

      13. add-sqr-sqrt 79.0%

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

   9. Applied egg-rr 79.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{+14} \lor \neg \left(t1 \leq 13000\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 6: 66.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.4 \cdot 10^{+47} \lor \neg \left(u \leq 8 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -9.4e+47) (not (<= u 8e+168)))
   (* (/ v u) (/ t1 u))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.4e+47) || !(u <= 8e+168)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -v / (t1 + 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 <= (-9.4d+47)) .or. (.not. (u <= 8d+168))) then
        tmp = (v / u) * (t1 / u)
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.4e+47) || !(u <= 8e+168)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -9.4e+47) or not (u <= 8e+168):
		tmp = (v / u) * (t1 / u)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -9.4e+47) || !(u <= 8e+168))
		tmp = Float64(Float64(v / u) * Float64(t1 / u));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -9.4e+47) || ~((u <= 8e+168)))
		tmp = (v / u) * (t1 / u);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -9.4e+47], N[Not[LessEqual[u, 8e+168]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -9.39999999999999928e47 or 7.9999999999999995e168 < u

   1. Initial program 78.6%

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

   2. Taylor expanded in t1 around 0 72.4%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{{u}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 72.4%

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

   4. Simplified 72.4%

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

      1. *-commutative 72.4%

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

      2. times-frac 81.9%

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

      3. add-sqr-sqrt 41.1%

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

      4. sqrt-unprod 59.4%

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

      5. sqr-neg 59.4%

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

      6. sqrt-unprod 32.9%

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

      7. add-sqr-sqrt 64.5%

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

   6. Applied egg-rr 64.5%

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

   if -9.39999999999999928e47 < u < 7.9999999999999995e168

   1. Initial program 72.2%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around inf 67.7%

      \[\leadsto \color{blue}{-1} \cdot \frac{v}{t1 + u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.4 \cdot 10^{+47} \lor \neg \left(u \leq 8 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 7: 66.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+48} \lor \neg \left(u \leq 1.45 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.9e+48) (not (<= u 1.45e+165)))
   (/ (/ t1 u) (/ u v))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.9e+48) || !(u <= 1.45e+165)) {
		tmp = (t1 / u) / (u / v);
	} else {
		tmp = -v / (t1 + 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 <= (-1.9d+48)) .or. (.not. (u <= 1.45d+165))) then
        tmp = (t1 / u) / (u / v)
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.9e+48) || !(u <= 1.45e+165)) {
		tmp = (t1 / u) / (u / v);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.9e+48) or not (u <= 1.45e+165):
		tmp = (t1 / u) / (u / v)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.9e+48) || !(u <= 1.45e+165))
		tmp = Float64(Float64(t1 / u) / Float64(u / v));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.9e+48) || ~((u <= 1.45e+165)))
		tmp = (t1 / u) / (u / v);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.9e+48], N[Not[LessEqual[u, 1.45e+165]], $MachinePrecision]], N[(N[(t1 / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.9e48 or 1.45000000000000003e165 < u

   1. Initial program 78.6%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in t1 around 0 81.9%

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

   5. Taylor expanded in t1 around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. neg-mul-1 81.9%

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

   7. Simplified 81.9%

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

      1. clear-num 81.8%

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

      2. un-div-inv 81.8%

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

      3. add-sqr-sqrt 40.9%

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

      4. sqrt-unprod 59.2%

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

      5. sqr-neg 59.2%

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

      6. sqrt-unprod 32.9%

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

      7. add-sqr-sqrt 64.5%

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

   9. Applied egg-rr 64.5%

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

   if -1.9e48 < u < 1.45000000000000003e165

   1. Initial program 72.2%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around inf 67.7%

      \[\leadsto \color{blue}{-1} \cdot \frac{v}{t1 + u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+48} \lor \neg \left(u \leq 1.45 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 8: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. *-commutative 74.2%

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

   2. times-frac 97.5%

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

   3. neg-mul-1 97.5%

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

   4. associate-/l* 97.2%

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

   5. associate-*r/ 97.2%

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

   6. associate-/l* 97.2%

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

   7. associate-/l/ 97.2%

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

   8. neg-mul-1 97.2%

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

   9. *-lft-identity 97.2%

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

   10. metadata-eval 97.2%

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

   11. times-frac 97.2%

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

   12. neg-mul-1 97.2%

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

   13. remove-double-neg 97.2%

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

   14. neg-mul-1 97.2%

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

   15. sub0-neg 97.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

   16. associate--r+ 97.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

   17. neg-sub0 97.2%

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

   18. div-sub 97.3%

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

   19. distribute-frac-neg 97.3%

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

   20. *-inverses 97.3%

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

   21. metadata-eval 97.3%

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

3. Simplified 97.3%

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

4. Final simplification 97.3%

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

Alternative 9: 55.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{v}{u} \cdot \left(-0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.2e+104) (* (/ v u) (- 0.5)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.2e+104) {
		tmp = (v / u) * -0.5;
	} 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 <= (-4.2d+104)) then
        tmp = (v / u) * -0.5d0
    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 <= -4.2e+104) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.2e+104:
		tmp = (v / u) * -0.5
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.2e+104)
		tmp = Float64(Float64(v / u) * Float64(-0.5));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.2e+104)
		tmp = (v / u) * -0.5;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.2e+104], N[(N[(v / u), $MachinePrecision] * (-0.5)), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.1999999999999997e104

   1. Initial program 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 93.7%

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

      3. neg-mul-1 93.7%

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

      4. associate-/l* 93.7%

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

      5. associate-*r/ 93.7%

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

      6. associate-/l* 93.7%

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

      7. associate-/l/ 93.7%

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

      8. neg-mul-1 93.7%

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

      9. *-lft-identity 93.7%

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

      10. metadata-eval 93.7%

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

      11. times-frac 93.7%

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

      12. neg-mul-1 93.7%

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

      13. remove-double-neg 93.7%

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

      14. neg-mul-1 93.7%

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

      15. sub0-neg 93.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 93.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 93.7%

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

      18. div-sub 93.7%

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

      19. distribute-frac-neg 93.7%

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

      20. *-inverses 93.7%

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

      21. metadata-eval 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in v around 0 82.8%

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

      1. mul-1-neg 82.8%

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

      2. +-commutative 82.8%

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

   6. Simplified 82.8%

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

   7. Taylor expanded in t1 around inf 44.2%

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

      1. *-commutative 44.2%

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

   9. Simplified 44.2%

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

   10. Taylor expanded in t1 around 0 33.5%

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

   if -4.1999999999999997e104 < u

   1. Initial program 73.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 61.3%

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

      1. associate-*r/ 61.3%

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

      2. neg-mul-1 61.3%

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

   6. Simplified 61.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{v}{u} \cdot \left(-0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10: 55.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.2e+107) (/ (- 0.5) (/ u v)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.2e+107) {
		tmp = -0.5 / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.2d+107)) then
        tmp = -0.5d0 / (u / v)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.2e+107) {
		tmp = -0.5 / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.2e+107:
		tmp = -0.5 / (u / v)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.2e+107)
		tmp = Float64(Float64(-0.5) / Float64(u / v));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.2e+107)
		tmp = -0.5 / (u / v);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.2e+107], N[((-0.5) / N[(u / v), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.2000000000000002e107

   1. Initial program 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 93.7%

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

      3. neg-mul-1 93.7%

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

      4. associate-/l* 93.7%

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

      5. associate-*r/ 93.7%

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

      6. associate-/l* 93.7%

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

      7. associate-/l/ 93.7%

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

      8. neg-mul-1 93.7%

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

      9. *-lft-identity 93.7%

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

      10. metadata-eval 93.7%

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

      11. times-frac 93.7%

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

      12. neg-mul-1 93.7%

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

      13. remove-double-neg 93.7%

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

      14. neg-mul-1 93.7%

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

      15. sub0-neg 93.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 93.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 93.7%

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

      18. div-sub 93.7%

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

      19. distribute-frac-neg 93.7%

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

      20. *-inverses 93.7%

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

      21. metadata-eval 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in v around 0 82.8%

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

      1. mul-1-neg 82.8%

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

      2. +-commutative 82.8%

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

   6. Simplified 82.8%

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

   7. Taylor expanded in t1 around inf 44.2%

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

      1. *-commutative 44.2%

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

   9. Simplified 44.2%

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

   10. Taylor expanded in t1 around 0 33.5%

       \[\leadsto -\color{blue}{0.5 \cdot \frac{v}{u}} \]
   11. Step-by-step derivation

       1. clear-num 33.9%

          \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{u}{v}}} \]

       2. un-div-inv 33.9%

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

   12. Applied egg-rr 33.9%

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

   if -5.2000000000000002e107 < u

   1. Initial program 73.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 61.3%

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

      1. associate-*r/ 61.3%

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

      2. neg-mul-1 61.3%

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

   6. Simplified 61.3%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 55.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+107}:\\ \;\;\;\;\frac{v \cdot \left(--0.5\right)}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.4e+107) (/ (* v (- -0.5)) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.4e+107) {
		tmp = (v * -(-0.5)) / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.4d+107)) then
        tmp = (v * -(-0.5d0)) / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.4e+107) {
		tmp = (v * -(-0.5)) / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.4e+107:
		tmp = (v * -(-0.5)) / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.4e+107)
		tmp = Float64(Float64(v * Float64(-(-0.5))) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.4e+107)
		tmp = (v * -(-0.5)) / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.4e+107], N[(N[(v * (--0.5)), $MachinePrecision] / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.3999999999999997e107

   1. Initial program 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 93.7%

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

      3. neg-mul-1 93.7%

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

      4. associate-/l* 93.7%

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

      5. associate-*r/ 93.7%

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

      6. associate-/l* 93.7%

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

      7. associate-/l/ 93.7%

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

      8. neg-mul-1 93.7%

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

      9. *-lft-identity 93.7%

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

      10. metadata-eval 93.7%

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

      11. times-frac 93.7%

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

      12. neg-mul-1 93.7%

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

      13. remove-double-neg 93.7%

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

      14. neg-mul-1 93.7%

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

      15. sub0-neg 93.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 93.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 93.7%

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

      18. div-sub 93.7%

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

      19. distribute-frac-neg 93.7%

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

      20. *-inverses 93.7%

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

      21. metadata-eval 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in v around 0 82.8%

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

      1. mul-1-neg 82.8%

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

      2. +-commutative 82.8%

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

   6. Simplified 82.8%

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

   7. Taylor expanded in t1 around inf 44.2%

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

      1. *-commutative 44.2%

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

   9. Simplified 44.2%

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

   10. Taylor expanded in t1 around 0 33.5%

       \[\leadsto -\color{blue}{0.5 \cdot \frac{v}{u}} \]
   11. Step-by-step derivation

       1. associate-*r/ 33.5%

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

       2. frac-2neg 33.5%

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

       3. add-sqr-sqrt 33.5%

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

       4. sqrt-unprod 58.6%

          \[\leadsto -\frac{-0.5 \cdot v}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}}} \]

       5. sqr-neg 58.6%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 34.1%

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

   12. Applied egg-rr 34.1%

       \[\leadsto -\color{blue}{\frac{-0.5 \cdot v}{u}} \]
   13. Step-by-step derivation

       1. distribute-lft-neg-in 34.1%

          \[\leadsto -\frac{\color{blue}{\left(-0.5\right) \cdot v}}{u} \]

       2. metadata-eval 34.1%

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

   14. Simplified 34.1%

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

   if -3.3999999999999997e107 < u

   1. Initial program 73.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 61.3%

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

      1. associate-*r/ 61.3%

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

      2. neg-mul-1 61.3%

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

   6. Simplified 61.3%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+107}:\\ \;\;\;\;\frac{v \cdot \left(--0.5\right)}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12: 55.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.25 \cdot 10^{+105}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.25e+105) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.25e+105) {
		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 <= (-2.25d+105)) 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 <= -2.25e+105) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.25e+105:
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.25e+105)
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.25e+105)
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.25e+105], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.2500000000000001e105

   1. Initial program 75.9%

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

      1. times-frac 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in t1 around 0 81.8%

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

   5. Taylor expanded in t1 around inf 33.5%

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

      1. associate-*r/ 33.5%

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

      2. neg-mul-1 33.5%

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

   7. Simplified 33.5%

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

   if -2.2500000000000001e105 < u

   1. Initial program 73.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 61.3%

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

      1. associate-*r/ 61.3%

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

      2. neg-mul-1 61.3%

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

   6. Simplified 61.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.25 \cdot 10^{+105}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 13: 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 74.2%

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

   1. times-frac 97.5%

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

3. Simplified 97.5%

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

4. Taylor expanded in t1 around inf 60.5%

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

5. Final simplification 60.5%

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

Alternative 14: 53.3% accurate, 3.0× 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(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 74.2%

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

   1. times-frac 97.5%

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

3. Simplified 97.5%

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

4. Taylor expanded in t1 around inf 54.4%

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

   1. associate-*r/ 54.4%

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

   2. neg-mul-1 54.4%

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

6. Simplified 54.4%

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

7. Final simplification 54.4%

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

Reproduce

herbie shell --seed 2023221 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))