Rosa's DopplerBench

Percentage Accurate: 72.8% → 98.2%

Time: 9.1s

Alternatives: 16

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: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. times-frac 98.0%

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

   2. distribute-frac-neg 98.0%

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

   3. distribute-neg-frac2 98.0%

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

   4. +-commutative 98.0%

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

   5. distribute-neg-in 98.0%

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

   6. unsub-neg 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 88.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+145} \lor \neg \left(t1 \leq 1.06 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.5e+145) (not (<= t1 1.06e+119)))
   (/ v (- (- t1) (* u 2.0)))
   (* v (/ t1 (* (+ t1 u) (- (- u) t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e+145) || !(t1 <= 1.06e+119)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = v * (t1 / ((t1 + u) * (-u - t1)));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4.5d+145)) .or. (.not. (t1 <= 1.06d+119))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = v * (t1 / ((t1 + u) * (-u - t1)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e+145) || !(t1 <= 1.06e+119)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = v * (t1 / ((t1 + u) * (-u - t1)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.5e+145) or not (t1 <= 1.06e+119):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = v * (t1 / ((t1 + u) * (-u - t1)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.5e+145) || !(t1 <= 1.06e+119))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) - t1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.5e+145) || ~((t1 <= 1.06e+119)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = v * (t1 / ((t1 + u) * (-u - t1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.5e+145], N[Not[LessEqual[t1, 1.06e+119]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.4999999999999998e145 or 1.0599999999999999e119 < t1

   1. Initial program 33.2%

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

      1. associate-*l/ 38.4%

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

      2. *-commutative 38.4%

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

   3. Simplified 38.4%

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

      1. associate-*r/ 33.2%

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

      2. *-commutative 33.2%

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

      3. associate-/r* 52.7%

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

      4. frac-2neg 52.7%

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

      5. distribute-lft-neg-out 52.7%

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

      6. remove-double-neg 52.7%

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

      7. +-commutative 52.7%

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

      8. distribute-neg-in 52.7%

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

      9. sub-neg 52.7%

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

      10. associate-*l/ 99.9%

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

      11. associate-*r/ 100.0%

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

      12. clear-num 100.0%

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

      13. frac-2neg 100.0%

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

      14. frac-times 98.8%

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

      15. *-un-lft-identity 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in u around 0 94.6%

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

      1. *-commutative 94.6%

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

   9. Simplified 94.6%

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

   if -4.4999999999999998e145 < t1 < 1.0599999999999999e119

   1. Initial program 83.9%

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

      1. associate-*l/ 87.8%

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

      2. *-commutative 87.8%

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

   3. Simplified 87.8%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+145} \lor \neg \left(t1 \leq 1.06 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.1 \cdot 10^{+19}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{-u}}{\frac{t1 + u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.1e+19)
   (* t1 (/ (/ v (+ t1 u)) (- u)))
   (if (<= u 7.2e+72) (/ v (- t1)) (/ (/ v (- u)) (/ (+ t1 u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.1e+19) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else if (u <= 7.2e+72) {
		tmp = v / -t1;
	} else {
		tmp = (v / -u) / ((t1 + u) / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.1d+19)) then
        tmp = t1 * ((v / (t1 + u)) / -u)
    else if (u <= 7.2d+72) then
        tmp = v / -t1
    else
        tmp = (v / -u) / ((t1 + u) / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.1e+19) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else if (u <= 7.2e+72) {
		tmp = v / -t1;
	} else {
		tmp = (v / -u) / ((t1 + u) / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.1e+19:
		tmp = t1 * ((v / (t1 + u)) / -u)
	elif u <= 7.2e+72:
		tmp = v / -t1
	else:
		tmp = (v / -u) / ((t1 + u) / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.1e+19)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(-u)));
	elseif (u <= 7.2e+72)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(Float64(v / Float64(-u)) / Float64(Float64(t1 + u) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.1e+19)
		tmp = t1 * ((v / (t1 + u)) / -u);
	elseif (u <= 7.2e+72)
		tmp = v / -t1;
	else
		tmp = (v / -u) / ((t1 + u) / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.1e+19], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.2e+72], N[(v / (-t1)), $MachinePrecision], N[(N[(v / (-u)), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.1e19

   1. Initial program 84.4%

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

      1. times-frac 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. distribute-neg-frac2 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-neg-in 98.5%

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

      6. unsub-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. mul-1-neg 78.8%

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

   7. Simplified 78.8%

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

      1. distribute-frac-neg 78.8%

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

      2. distribute-lft-neg-out 78.8%

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

      3. add-sqr-sqrt 32.4%

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

      4. sqrt-unprod 54.9%

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

      5. sqr-neg 54.9%

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

      6. sqrt-unprod 33.6%

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

      7. add-sqr-sqrt 57.9%

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

      8. frac-2neg 57.9%

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

      9. clear-num 57.9%

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

      10. frac-times 55.4%

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

      11. *-un-lft-identity 55.4%

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

      12. add-sqr-sqrt 32.9%

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

      13. sqrt-unprod 57.7%

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

      14. sqr-neg 57.7%

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

      15. sqrt-unprod 30.7%

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

      16. add-sqr-sqrt 70.7%

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

      17. add-sqr-sqrt 42.2%

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

      18. sqrt-unprod 54.9%

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

      19. sqr-neg 54.9%

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

      20. sqrt-unprod 21.7%

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

      21. add-sqr-sqrt 55.4%

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

   9. Applied egg-rr 70.7%

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

       1. associate-/l/ 78.8%

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

       2. associate-/r/ 79.1%

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

       3. distribute-rgt-neg-in 79.1%

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

   11. Simplified 79.1%

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

   if -1.1e19 < u < 7.20000000000000069e72

   1. Initial program 56.6%

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

      1. associate-*l/ 67.3%

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

      2. *-commutative 67.3%

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

   3. Simplified 67.3%

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

   5. Taylor expanded in t1 around inf 78.9%

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

      1. associate-*r/ 78.9%

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

      2. neg-mul-1 78.9%

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

   7. Simplified 78.9%

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

   if 7.20000000000000069e72 < u

   1. Initial program 76.7%

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

      1. associate-*l/ 72.8%

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

      2. *-commutative 72.8%

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

   3. Simplified 72.8%

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

      1. associate-*r/ 76.7%

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

      2. *-commutative 76.7%

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

      3. associate-/r* 93.1%

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

      4. frac-2neg 93.1%

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

      5. distribute-lft-neg-out 93.1%

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

      6. remove-double-neg 93.1%

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

      7. +-commutative 93.1%

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

      8. distribute-neg-in 93.1%

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

      9. sub-neg 93.1%

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

      10. associate-*l/ 97.7%

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

      11. associate-*r/ 97.8%

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

      12. clear-num 97.7%

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

      13. frac-2neg 97.7%

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

      14. frac-times 85.4%

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

      15. *-un-lft-identity 85.4%

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

   6. Applied egg-rr 85.4%

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

   7. Taylor expanded in t1 around 0 79.3%

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

      1. neg-mul-1 79.3%

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

      2. times-frac 91.6%

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

      3. +-commutative 91.6%

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

   9. Applied egg-rr 91.6%

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

       1. associate-*l/ 91.8%

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

       2. associate-*r/ 91.8%

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

       3. mul-1-neg 91.8%

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

   11. Simplified 91.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.1 \cdot 10^{+19}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{-u}}{\frac{t1 + u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-44} \lor \neg \left(t1 \leq 1.22 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.2e-44) (not (<= t1 1.22e+66)))
   (/ v (- (- t1) (* u 2.0)))
   (* (/ v (- u)) (/ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.2e-44) || !(t1 <= 1.22e+66)) {
		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 <= (-2.2d-44)) .or. (.not. (t1 <= 1.22d+66))) 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 <= -2.2e-44) || !(t1 <= 1.22e+66)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / -u) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.2e-44) or not (t1 <= 1.22e+66):
		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 <= -2.2e-44) || !(t1 <= 1.22e+66))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v / Float64(-u)) * Float64(t1 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.2e-44) || ~((t1 <= 1.22e+66)))
		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, -2.2e-44], N[Not[LessEqual[t1, 1.22e+66]], $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 < -2.20000000000000012e-44 or 1.21999999999999993e66 < t1

   1. Initial program 53.9%

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

      1. associate-*l/ 57.1%

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

      2. *-commutative 57.1%

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

   3. Simplified 57.1%

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

      1. associate-*r/ 53.9%

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

      2. *-commutative 53.9%

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

      3. associate-/r* 70.3%

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

      4. frac-2neg 70.3%

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

      5. distribute-lft-neg-out 70.3%

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

      6. remove-double-neg 70.3%

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

      7. +-commutative 70.3%

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

      8. distribute-neg-in 70.3%

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

      9. sub-neg 70.3%

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

      10. associate-*l/ 99.9%

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

      11. associate-*r/ 100.0%

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

      12. clear-num 99.9%

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

      13. frac-2neg 99.9%

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

      14. frac-times 96.4%

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

      15. *-un-lft-identity 96.4%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in u around 0 86.1%

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

      1. *-commutative 86.1%

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

   9. Simplified 86.1%

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

   if -2.20000000000000012e-44 < t1 < 1.21999999999999993e66

   1. Initial program 82.1%

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

      1. times-frac 96.1%

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

      2. distribute-frac-neg 96.1%

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

      3. distribute-neg-frac2 96.1%

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

      4. +-commutative 96.1%

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

      5. distribute-neg-in 96.1%

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

      6. unsub-neg 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in t1 around 0 73.3%

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

      1. associate-*r/ 73.3%

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

      2. mul-1-neg 73.3%

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

   7. Simplified 73.3%

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

   8. Taylor expanded in t1 around 0 76.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-44} \lor \neg \left(t1 \leq 1.22 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{-43} \lor \neg \left(t1 \leq 5 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.5e-43) (not (<= t1 5e+66)))
   (/ v (- (- u) t1))
   (* (/ v (- u)) (/ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e-43) || !(t1 <= 5e+66)) {
		tmp = v / (-u - t1);
	} 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 <= (-1.5d-43)) .or. (.not. (t1 <= 5d+66))) then
        tmp = v / (-u - t1)
    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 <= -1.5e-43) || !(t1 <= 5e+66)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v / -u) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.5e-43) or not (t1 <= 5e+66):
		tmp = v / (-u - t1)
	else:
		tmp = (v / -u) * (t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.5e-43) || !(t1 <= 5e+66))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(v / Float64(-u)) * Float64(t1 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.5e-43) || ~((t1 <= 5e+66)))
		tmp = v / (-u - t1);
	else
		tmp = (v / -u) * (t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.5e-43], N[Not[LessEqual[t1, 5e+66]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / (-u)), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.50000000000000002e-43 or 4.99999999999999991e66 < t1

   1. Initial program 53.9%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 85.2%

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

      1. add-sqr-sqrt 33.3%

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

      2. sqrt-unprod 86.7%

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

      3. sqr-neg 86.7%

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

      4. sqrt-unprod 51.8%

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

      5. add-sqr-sqrt 85.1%

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

      6. sub-neg 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. mul-1-neg 85.1%

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

      2. neg-sub0 85.1%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]

   9. Applied egg-rr 85.1%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]
   10. Step-by-step derivation

       1. neg-sub0 85.1%

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

       2. distribute-neg-frac 85.1%

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

   11. Simplified 85.1%

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

       1. distribute-frac-neg 85.1%

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

       2. neg-sub0 85.1%

          \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]

       3. sub-neg 85.1%

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

       4. add-sqr-sqrt 36.6%

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

       5. sqrt-unprod 32.8%

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

       6. sqr-neg 32.8%

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

       7. sqrt-unprod 14.6%

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

       8. add-sqr-sqrt 25.1%

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

       9. distribute-neg-in 25.1%

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

       10. remove-double-neg 25.1%

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

       11. frac-2neg 25.1%

           \[\leadsto 0 - \color{blue}{\frac{-v}{t1 + u}} \]

       12. add-sqr-sqrt 10.6%

           \[\leadsto 0 - \frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1 + u} \]

       13. sqrt-unprod 41.5%

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

       14. sqr-neg 41.5%

           \[\leadsto 0 - \frac{\sqrt{\color{blue}{v \cdot v}}}{t1 + u} \]

       15. sqrt-unprod 41.8%

           \[\leadsto 0 - \frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1 + u} \]

       16. add-sqr-sqrt 85.2%

           \[\leadsto 0 - \frac{\color{blue}{v}}{t1 + u} \]

   13. Applied egg-rr 85.2%

       \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]
   14. Step-by-step derivation

       1. neg-sub0 85.2%

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

       2. distribute-neg-frac2 85.2%

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

   15. Simplified 85.2%

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

   if -1.50000000000000002e-43 < t1 < 4.99999999999999991e66

   1. Initial program 82.1%

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

      1. times-frac 96.1%

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

      2. distribute-frac-neg 96.1%

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

      3. distribute-neg-frac2 96.1%

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

      4. +-commutative 96.1%

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

      5. distribute-neg-in 96.1%

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

      6. unsub-neg 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in t1 around 0 73.3%

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

      1. associate-*r/ 73.3%

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

      2. mul-1-neg 73.3%

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

   7. Simplified 73.3%

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

   8. Taylor expanded in t1 around 0 76.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{-43} \lor \neg \left(t1 \leq 5 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.45e+20)
   (/ t1 (* u (/ (- u) v)))
   (if (<= u 7.2e+72) (/ v (- t1)) (* (/ v (- u)) (/ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.45e+20) {
		tmp = t1 / (u * (-u / v));
	} else if (u <= 7.2e+72) {
		tmp = v / -t1;
	} 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 (u <= (-2.45d+20)) then
        tmp = t1 / (u * (-u / v))
    else if (u <= 7.2d+72) then
        tmp = v / -t1
    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 (u <= -2.45e+20) {
		tmp = t1 / (u * (-u / v));
	} else if (u <= 7.2e+72) {
		tmp = v / -t1;
	} else {
		tmp = (v / -u) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.45e+20:
		tmp = t1 / (u * (-u / v))
	elif u <= 7.2e+72:
		tmp = v / -t1
	else:
		tmp = (v / -u) * (t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.45e+20)
		tmp = Float64(t1 / Float64(u * Float64(Float64(-u) / v)));
	elseif (u <= 7.2e+72)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(Float64(v / Float64(-u)) * Float64(t1 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.45e+20)
		tmp = t1 / (u * (-u / v));
	elseif (u <= 7.2e+72)
		tmp = v / -t1;
	else
		tmp = (v / -u) * (t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.45e+20], N[(t1 / N[(u * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.2e+72], N[(v / (-t1)), $MachinePrecision], N[(N[(v / (-u)), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.45e20

   1. Initial program 84.4%

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

      1. times-frac 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. distribute-neg-frac2 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-neg-in 98.5%

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

      6. unsub-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. mul-1-neg 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in t1 around 0 76.1%

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

      1. *-commutative 76.1%

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

      2. clear-num 76.0%

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

      3. frac-2neg 76.0%

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

      4. frac-times 78.7%

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

      5. *-un-lft-identity 78.7%

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

      6. remove-double-neg 78.7%

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

   10. Applied egg-rr 78.7%

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

   if -2.45e20 < u < 7.20000000000000069e72

   1. Initial program 56.6%

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

      1. associate-*l/ 67.3%

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

      2. *-commutative 67.3%

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

   3. Simplified 67.3%

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

   5. Taylor expanded in t1 around inf 78.9%

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

      1. associate-*r/ 78.9%

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

      2. neg-mul-1 78.9%

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

   7. Simplified 78.9%

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

   if 7.20000000000000069e72 < u

   1. Initial program 76.7%

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

      1. times-frac 97.8%

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

      2. distribute-frac-neg 97.8%

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

      3. distribute-neg-frac2 97.8%

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

      4. +-commutative 97.8%

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

      5. distribute-neg-in 97.8%

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

      6. unsub-neg 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in t1 around 0 91.8%

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

      1. associate-*r/ 91.8%

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

      2. mul-1-neg 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in t1 around 0 91.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+86} \lor \neg \left(u \leq 3.5 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.8e+86) (not (<= u 3.5e+73)))
   (/ t1 (* u (/ u v)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.8e+86) || !(u <= 3.5e+73)) {
		tmp = t1 / (u * (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 <= (-3.8d+86)) .or. (.not. (u <= 3.5d+73))) then
        tmp = t1 / (u * (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 <= -3.8e+86) || !(u <= 3.5e+73)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.8e+86) or not (u <= 3.5e+73):
		tmp = t1 / (u * (u / v))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.8e+86) || !(u <= 3.5e+73))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.8e+86) || ~((u <= 3.5e+73)))
		tmp = t1 / (u * (u / v));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.8e+86], N[Not[LessEqual[u, 3.5e+73]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.79999999999999978e86 or 3.50000000000000002e73 < u

   1. Initial program 80.5%

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

      1. times-frac 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. distribute-neg-frac2 99.0%

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

      4. +-commutative 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. unsub-neg 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in t1 around 0 89.0%

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

      1. associate-*r/ 89.0%

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

      2. mul-1-neg 89.0%

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

   7. Simplified 89.0%

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

   8. Taylor expanded in t1 around 0 87.0%

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

      1. *-commutative 87.0%

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

      2. clear-num 86.9%

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

      3. frac-times 86.6%

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

      4. *-un-lft-identity 86.6%

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

      5. add-sqr-sqrt 44.5%

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

      6. sqrt-unprod 66.6%

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

      7. sqr-neg 66.6%

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

      8. sqrt-unprod 33.1%

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

      9. add-sqr-sqrt 71.3%

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

   10. Applied egg-rr 71.3%

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

   if -3.79999999999999978e86 < u < 3.50000000000000002e73

   1. Initial program 60.3%

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

      1. associate-*l/ 68.0%

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

      2. *-commutative 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in t1 around inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. neg-mul-1 73.2%

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

   7. Simplified 73.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+86} \lor \neg \left(u \leq 3.5 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+73}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.2e+86)
   (/ t1 (* u (/ u v)))
   (if (<= u 3.4e+73) (/ v (- t1)) (/ v (* u (/ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.2e+86) {
		tmp = t1 / (u * (u / v));
	} else if (u <= 3.4e+73) {
		tmp = v / -t1;
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.2d+86)) then
        tmp = t1 / (u * (u / v))
    else if (u <= 3.4d+73) then
        tmp = v / -t1
    else
        tmp = v / (u * (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.2e+86) {
		tmp = t1 / (u * (u / v));
	} else if (u <= 3.4e+73) {
		tmp = v / -t1;
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.2e+86:
		tmp = t1 / (u * (u / v))
	elif u <= 3.4e+73:
		tmp = v / -t1
	else:
		tmp = v / (u * (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.2e+86)
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	elseif (u <= 3.4e+73)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.2e+86)
		tmp = t1 / (u * (u / v));
	elseif (u <= 3.4e+73)
		tmp = v / -t1;
	else
		tmp = v / (u * (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.2e+86], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.4e+73], N[(v / (-t1)), $MachinePrecision], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.2e86

   1. Initial program 84.0%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around 0 86.5%

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

      1. associate-*r/ 86.5%

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

      2. mul-1-neg 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in t1 around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. clear-num 82.7%

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

      3. frac-times 86.4%

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

      4. *-un-lft-identity 86.4%

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

      5. add-sqr-sqrt 49.4%

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

      6. sqrt-unprod 64.0%

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

      7. sqr-neg 64.0%

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

      8. sqrt-unprod 32.9%

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

      9. add-sqr-sqrt 78.2%

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

   10. Applied egg-rr 78.2%

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

   if -3.2e86 < u < 3.4000000000000002e73

   1. Initial program 60.3%

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

      1. associate-*l/ 68.0%

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

      2. *-commutative 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in t1 around inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. neg-mul-1 73.2%

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

   7. Simplified 73.2%

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

   if 3.4000000000000002e73 < u

   1. Initial program 76.7%

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

      1. times-frac 97.8%

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

      2. distribute-frac-neg 97.8%

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

      3. distribute-neg-frac2 97.8%

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

      4. +-commutative 97.8%

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

      5. distribute-neg-in 97.8%

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

      6. unsub-neg 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in t1 around 0 91.8%

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

      1. associate-*r/ 91.8%

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

      2. mul-1-neg 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in t1 around 0 91.7%

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

      1. clear-num 91.5%

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

      2. frac-times 79.2%

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

      3. *-un-lft-identity 79.2%

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

      4. add-sqr-sqrt 34.8%

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

      5. sqrt-unprod 65.3%

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

      6. sqr-neg 65.3%

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

      7. sqrt-unprod 33.4%

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

      8. add-sqr-sqrt 63.8%

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

   10. Applied egg-rr 63.8%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+73}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+118} \lor \neg \left(u \leq 1.7 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.2e+118) (not (<= u 1.7e+123))) (/ 1.0 (/ u v)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.2e+118) || !(u <= 1.7e+123)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4.2d+118)) .or. (.not. (u <= 1.7d+123))) then
        tmp = 1.0d0 / (u / v)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.2e+118) || !(u <= 1.7e+123)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.2e+118) or not (u <= 1.7e+123):
		tmp = 1.0 / (u / v)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.2e+118) || !(u <= 1.7e+123))
		tmp = Float64(1.0 / Float64(u / v));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.2e+118) || ~((u <= 1.7e+123)))
		tmp = 1.0 / (u / v);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.2e+118], N[Not[LessEqual[u, 1.7e+123]], $MachinePrecision]], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.2e118 or 1.70000000000000001e123 < u

   1. Initial program 79.1%

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

      1. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around inf 50.6%

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

      1. add-sqr-sqrt 20.2%

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

      2. sqrt-unprod 73.7%

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

      3. sqr-neg 73.7%

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

      4. sqrt-unprod 30.2%

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

      5. add-sqr-sqrt 50.5%

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

      6. sub-neg 50.5%

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

   7. Applied egg-rr 50.5%

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

   8. Taylor expanded in t1 around 0 43.9%

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

      1. clear-num 45.9%

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

      2. inv-pow 45.9%

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

   10. Applied egg-rr 45.9%

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

       1. unpow-1 45.9%

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

   12. Simplified 45.9%

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

   if -4.2e118 < u < 1.70000000000000001e123

   1. Initial program 62.7%

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

      1. associate-*l/ 69.2%

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

      2. *-commutative 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in t1 around inf 69.7%

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

      1. associate-*r/ 69.7%

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

      2. neg-mul-1 69.7%

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

   7. Simplified 69.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+118} \lor \neg \left(u \leq 1.7 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+118} \lor \neg \left(u \leq 4.4 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.2e+118) (not (<= u 4.4e+73))) (* (/ v u) -0.5) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.2e+118) || !(u <= 4.4e+73)) {
		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 <= (-6.2d+118)) .or. (.not. (u <= 4.4d+73))) 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 <= -6.2e+118) || !(u <= 4.4e+73)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.2e+118) or not (u <= 4.4e+73):
		tmp = (v / u) * -0.5
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.2e+118) || !(u <= 4.4e+73))
		tmp = Float64(Float64(v / u) * -0.5);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.2e+118) || ~((u <= 4.4e+73)))
		tmp = (v / u) * -0.5;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.2e+118], N[Not[LessEqual[u, 4.4e+73]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.19999999999999973e118 or 4.4e73 < u

   1. Initial program 80.4%

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

      1. associate-*l/ 78.4%

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

      2. *-commutative 78.4%

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

   3. Simplified 78.4%

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

      1. associate-*r/ 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-/r* 92.7%

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

      4. frac-2neg 92.7%

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

      5. distribute-lft-neg-out 92.7%

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

      6. remove-double-neg 92.7%

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

      7. +-commutative 92.7%

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

      8. distribute-neg-in 92.7%

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

      9. sub-neg 92.7%

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

      10. associate-*l/ 98.8%

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

      11. associate-*r/ 98.8%

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

      12. clear-num 98.7%

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

      13. frac-2neg 98.7%

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

      14. frac-times 89.8%

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

      15. *-un-lft-identity 89.8%

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

   6. Applied egg-rr 89.8%

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

   7. Taylor expanded in u around 0 48.3%

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

      1. *-commutative 48.3%

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

   9. Simplified 48.3%

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

   10. Taylor expanded in t1 around 0 41.7%

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

       1. *-commutative 41.7%

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

   12. Simplified 41.7%

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

   if -6.19999999999999973e118 < u < 4.4e73

   1. Initial program 61.7%

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

      1. associate-*l/ 68.9%

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

      2. *-commutative 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in t1 around inf 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   7. Simplified 71.6%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+118} \lor \neg \left(u \leq 4.4 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+117} \lor \neg \left(u \leq 4.4 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.4e+117) (not (<= u 4.4e+73))) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+117) || !(u <= 4.4e+73)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.4d+117)) .or. (.not. (u <= 4.4d+73))) then
        tmp = v / -u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+117) || !(u <= 4.4e+73)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.4e+117) or not (u <= 4.4e+73):
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.4e+117) || !(u <= 4.4e+73))
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.4e+117) || ~((u <= 4.4e+73)))
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.4e+117], N[Not[LessEqual[u, 4.4e+73]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.4000000000000001e117 or 4.4e73 < u

   1. Initial program 80.4%

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

      1. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around inf 47.8%

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

   6. Taylor expanded in t1 around 0 41.6%

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

      1. associate-*r/ 41.6%

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

      2. mul-1-neg 41.6%

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

   8. Simplified 41.6%

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

   if -3.4000000000000001e117 < u < 4.4e73

   1. Initial program 61.7%

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

      1. associate-*l/ 68.9%

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

      2. *-commutative 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in t1 around inf 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   7. Simplified 71.6%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+117} \lor \neg \left(u \leq 4.4 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.86 \cdot 10^{+118} \lor \neg \left(u \leq 1.2 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.86e+118) (not (<= u 1.2e+116))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.86e+118) || !(u <= 1.2e+116)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.86d+118)) .or. (.not. (u <= 1.2d+116))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.86e+118) || !(u <= 1.2e+116)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.86e+118) or not (u <= 1.2e+116):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.86e+118) || !(u <= 1.2e+116))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.86e+118) || ~((u <= 1.2e+116)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.86e+118], N[Not[LessEqual[u, 1.2e+116]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.85999999999999996e118 or 1.2e116 < u

   1. Initial program 79.1%

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

      1. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around inf 50.6%

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

      1. add-sqr-sqrt 20.2%

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

      2. sqrt-unprod 73.7%

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

      3. sqr-neg 73.7%

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

      4. sqrt-unprod 30.2%

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

      5. add-sqr-sqrt 50.5%

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

      6. sub-neg 50.5%

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

   7. Applied egg-rr 50.5%

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

   8. Taylor expanded in t1 around 0 43.9%

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

   if -1.85999999999999996e118 < u < 1.2e116

   1. Initial program 62.7%

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

      1. associate-*l/ 69.2%

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

      2. *-commutative 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in t1 around inf 69.7%

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

      1. associate-*r/ 69.7%

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

      2. neg-mul-1 69.7%

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

   7. Simplified 69.7%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.86 \cdot 10^{+118} \lor \neg \left(u \leq 1.2 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 24.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{+66} \lor \neg \left(t1 \leq 4.9 \cdot 10^{+117}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.5e+66) (not (<= t1 4.9e+117))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e+66) || !(t1 <= 4.9e+117)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.5d+66)) .or. (.not. (t1 <= 4.9d+117))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e+66) || !(t1 <= 4.9e+117)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.5e+66) or not (t1 <= 4.9e+117):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.5e+66) || !(t1 <= 4.9e+117))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.5e+66) || ~((t1 <= 4.9e+117)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.5e+66], N[Not[LessEqual[t1, 4.9e+117]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.50000000000000001e66 or 4.9000000000000001e117 < t1

   1. Initial program 43.0%

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

      1. associate-*l/ 47.2%

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

      2. *-commutative 47.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in t1 around inf 89.3%

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

      1. associate-*r/ 89.3%

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

      2. neg-mul-1 89.3%

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

   7. Simplified 89.3%

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

      1. neg-sub0 89.3%

         \[\leadsto \frac{\color{blue}{0 - v}}{t1} \]

      2. sub-neg 89.3%

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

      3. add-sqr-sqrt 44.6%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1} \]

      4. sqrt-unprod 45.9%

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

      5. sqr-neg 45.9%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{v \cdot v}}}{t1} \]

      6. sqrt-unprod 14.5%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1} \]

      7. add-sqr-sqrt 24.8%

         \[\leadsto \frac{0 + \color{blue}{v}}{t1} \]

   9. Applied egg-rr 24.8%

      \[\leadsto \frac{\color{blue}{0 + v}}{t1} \]
   10. Step-by-step derivation

       1. +-lft-identity 24.8%

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

   11. Simplified 24.8%

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

   if -1.50000000000000001e66 < t1 < 4.9000000000000001e117

   1. Initial program 83.7%

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

      1. times-frac 96.8%

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

      2. distribute-frac-neg 96.8%

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

      3. distribute-neg-frac2 96.8%

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

      4. +-commutative 96.8%

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

      5. distribute-neg-in 96.8%

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

      6. unsub-neg 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in t1 around inf 46.2%

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

      1. add-sqr-sqrt 21.6%

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

      2. sqrt-unprod 58.8%

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

      3. sqr-neg 58.8%

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

      4. sqrt-unprod 24.4%

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

      5. add-sqr-sqrt 46.5%

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

      6. sub-neg 46.5%

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

   7. Applied egg-rr 46.5%

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

   8. Taylor expanded in t1 around 0 22.7%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{+66} \lor \neg \left(t1 \leq 4.9 \cdot 10^{+117}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. times-frac 98.0%

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

   2. distribute-frac-neg 98.0%

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

   3. distribute-neg-frac2 98.0%

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

   4. +-commutative 98.0%

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

   5. distribute-neg-in 98.0%

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

   6. unsub-neg 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in t1 around inf 64.0%

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

   1. add-sqr-sqrt 26.9%

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

   2. sqrt-unprod 71.3%

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

   3. sqr-neg 71.3%

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

   4. sqrt-unprod 36.9%

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

   5. add-sqr-sqrt 64.1%

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

   6. sub-neg 64.1%

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

7. Applied egg-rr 64.1%

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

   1. mul-1-neg 64.1%

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

   2. neg-sub0 64.1%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]

9. Applied egg-rr 64.1%

   \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]
10. Step-by-step derivation

    1. neg-sub0 64.1%

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

    2. distribute-neg-frac 64.1%

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

11. Simplified 64.1%

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

    1. distribute-frac-neg 64.1%

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

    2. neg-sub0 64.1%

       \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]

    3. sub-neg 64.1%

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

    4. add-sqr-sqrt 30.3%

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

    5. sqrt-unprod 32.7%

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

    6. sqr-neg 32.7%

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

    7. sqrt-unprod 12.0%

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

    8. add-sqr-sqrt 22.8%

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

    9. distribute-neg-in 22.8%

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

    10. remove-double-neg 22.8%

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

    11. frac-2neg 22.8%

        \[\leadsto 0 - \color{blue}{\frac{-v}{t1 + u}} \]

    12. add-sqr-sqrt 11.7%

        \[\leadsto 0 - \frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1 + u} \]

    13. sqrt-unprod 36.1%

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

    14. sqr-neg 36.1%

        \[\leadsto 0 - \frac{\sqrt{\color{blue}{v \cdot v}}}{t1 + u} \]

    15. sqrt-unprod 30.6%

        \[\leadsto 0 - \frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1 + u} \]

    16. add-sqr-sqrt 64.0%

        \[\leadsto 0 - \frac{\color{blue}{v}}{t1 + u} \]

13. Applied egg-rr 64.0%

    \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]
14. Step-by-step derivation

    1. neg-sub0 64.0%

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

    2. distribute-neg-frac2 64.0%

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

15. Simplified 64.0%

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

16. Final simplification 64.0%

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

Alternative 15: 62.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. times-frac 98.0%

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

   2. distribute-frac-neg 98.0%

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

   3. distribute-neg-frac2 98.0%

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

   4. +-commutative 98.0%

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

   5. distribute-neg-in 98.0%

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

   6. unsub-neg 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in t1 around inf 64.0%

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

   1. add-sqr-sqrt 26.9%

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

   2. sqrt-unprod 71.3%

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

   3. sqr-neg 71.3%

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

   4. sqrt-unprod 36.9%

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

   5. add-sqr-sqrt 64.1%

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

   6. sub-neg 64.1%

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

7. Applied egg-rr 64.1%

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

   1. mul-1-neg 64.1%

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

   2. neg-sub0 64.1%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]

9. Applied egg-rr 64.1%

   \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]
10. Step-by-step derivation

    1. neg-sub0 64.1%

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

    2. distribute-neg-frac 64.1%

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

11. Simplified 64.1%

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

12. Final simplification 64.1%

    \[\leadsto \frac{v}{u - t1} \]
13. Add Preprocessing

Alternative 16: 14.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. associate-*l/ 72.0%

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

   2. *-commutative 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in t1 around inf 54.7%

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

   1. associate-*r/ 54.7%

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

   2. neg-mul-1 54.7%

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

7. Simplified 54.7%

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

   1. neg-sub0 54.7%

      \[\leadsto \frac{\color{blue}{0 - v}}{t1} \]

   2. sub-neg 54.7%

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

   3. add-sqr-sqrt 27.2%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1} \]

   4. sqrt-unprod 36.0%

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

   5. sqr-neg 36.0%

      \[\leadsto \frac{0 + \sqrt{\color{blue}{v \cdot v}}}{t1} \]

   6. sqrt-unprod 6.7%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1} \]

   7. add-sqr-sqrt 11.6%

      \[\leadsto \frac{0 + \color{blue}{v}}{t1} \]

9. Applied egg-rr 11.6%

   \[\leadsto \frac{\color{blue}{0 + v}}{t1} \]
10. Step-by-step derivation

    1. +-lft-identity 11.6%

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

11. Simplified 11.6%

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

Reproduce

herbie shell --seed 2024180 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))