Rosa's DopplerBench

Percentage Accurate: 73.1% → 98.2%

Time: 12.1s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. times-frac 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t1 - u}{v \cdot \frac{t1}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.2e-101)
   (/ t1 (* (/ u v) (- t1 u)))
   (if (<= u 2.3e-48) (/ (- v) t1) (/ 1.0 (/ (- t1 u) (* v (/ t1 u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.2e-101) {
		tmp = t1 / ((u / v) * (t1 - u));
	} else if (u <= 2.3e-48) {
		tmp = -v / t1;
	} else {
		tmp = 1.0 / ((t1 - u) / (v * (t1 / u)));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.2d-101)) then
        tmp = t1 / ((u / v) * (t1 - u))
    else if (u <= 2.3d-48) then
        tmp = -v / t1
    else
        tmp = 1.0d0 / ((t1 - u) / (v * (t1 / u)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.2e-101) {
		tmp = t1 / ((u / v) * (t1 - u));
	} else if (u <= 2.3e-48) {
		tmp = -v / t1;
	} else {
		tmp = 1.0 / ((t1 - u) / (v * (t1 / u)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.2e-101:
		tmp = t1 / ((u / v) * (t1 - u))
	elif u <= 2.3e-48:
		tmp = -v / t1
	else:
		tmp = 1.0 / ((t1 - u) / (v * (t1 / u)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.2e-101)
		tmp = Float64(t1 / Float64(Float64(u / v) * Float64(t1 - u)));
	elseif (u <= 2.3e-48)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(1.0 / Float64(Float64(t1 - u) / Float64(v * Float64(t1 / u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.2e-101)
		tmp = t1 / ((u / v) * (t1 - u));
	elseif (u <= 2.3e-48)
		tmp = -v / t1;
	else
		tmp = 1.0 / ((t1 - u) / (v * (t1 / u)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.2e-101], N[(t1 / N[(N[(u / v), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.3e-48], N[((-v) / t1), $MachinePrecision], N[(1.0 / N[(N[(t1 - u), $MachinePrecision] / N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.1999999999999999e-101

   1. Initial program 75.7%

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

      1. times-frac 98.6%

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

   3. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. clear-num 98.6%

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

      3. frac-2neg 98.6%

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

      4. frac-times 85.1%

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

      5. *-un-lft-identity 85.1%

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

      6. remove-double-neg 85.1%

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

      7. distribute-neg-in 85.1%

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

      8. add-sqr-sqrt 51.1%

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

      9. sqrt-unprod 79.5%

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

      10. sqr-neg 79.5%

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

      11. sqrt-unprod 30.6%

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

      12. add-sqr-sqrt 76.4%

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

      13. sub-neg 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in t1 around 0 77.5%

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

   if -2.1999999999999999e-101 < u < 2.3000000000000001e-48

   1. Initial program 62.5%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in t1 around inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   7. Simplified 86.9%

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

   if 2.3000000000000001e-48 < u

   1. Initial program 76.0%

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

      1. times-frac 96.5%

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

   3. Simplified 96.5%

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

      1. *-commutative 96.5%

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

      2. clear-num 95.3%

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

      3. frac-2neg 95.3%

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

      4. frac-times 86.0%

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

      5. *-un-lft-identity 86.0%

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

      6. remove-double-neg 86.0%

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

      7. distribute-neg-in 86.0%

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

      8. add-sqr-sqrt 45.3%

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

      9. sqrt-unprod 78.1%

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

      10. sqr-neg 78.1%

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

      11. sqrt-unprod 34.2%

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

      12. add-sqr-sqrt 73.6%

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

      13. sub-neg 73.6%

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

   6. Applied egg-rr 73.6%

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

   7. Taylor expanded in t1 around 0 73.6%

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

      1. clear-num 73.7%

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

      2. inv-pow 73.7%

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

      3. *-commutative 73.7%

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

      4. clear-num 73.7%

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

      5. un-div-inv 73.6%

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

   9. Applied egg-rr 73.6%

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

       1. associate-/l/ 78.0%

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

       2. associate-*r/ 74.9%

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

       3. *-commutative 74.9%

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

       4. associate-*r/ 78.0%

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

       5. unpow-1 78.0%

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

   11. Simplified 78.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t1 - u}{v \cdot \frac{t1}{u}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{-36}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.6e-101)
   (/ t1 (* (/ u v) (- t1 u)))
   (if (<= u 6e-36) (/ (- v) t1) (* (/ (- t1) (+ t1 u)) (/ v u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e-101) {
		tmp = t1 / ((u / v) * (t1 - u));
	} else if (u <= 6e-36) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / (t1 + u)) * (v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.6d-101)) then
        tmp = t1 / ((u / v) * (t1 - u))
    else if (u <= 6d-36) then
        tmp = -v / t1
    else
        tmp = (-t1 / (t1 + u)) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e-101) {
		tmp = t1 / ((u / v) * (t1 - u));
	} else if (u <= 6e-36) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / (t1 + u)) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.6e-101:
		tmp = t1 / ((u / v) * (t1 - u))
	elif u <= 6e-36:
		tmp = -v / t1
	else:
		tmp = (-t1 / (t1 + u)) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.6e-101)
		tmp = Float64(t1 / Float64(Float64(u / v) * Float64(t1 - u)));
	elseif (u <= 6e-36)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(t1 + u)) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.6e-101)
		tmp = t1 / ((u / v) * (t1 - u));
	elseif (u <= 6e-36)
		tmp = -v / t1;
	else
		tmp = (-t1 / (t1 + u)) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.6e-101], N[(t1 / N[(N[(u / v), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 6e-36], N[((-v) / t1), $MachinePrecision], N[(N[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.6000000000000001e-101

   1. Initial program 75.7%

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

      1. times-frac 98.6%

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

   3. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. clear-num 98.6%

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

      3. frac-2neg 98.6%

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

      4. frac-times 85.1%

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

      5. *-un-lft-identity 85.1%

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

      6. remove-double-neg 85.1%

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

      7. distribute-neg-in 85.1%

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

      8. add-sqr-sqrt 51.1%

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

      9. sqrt-unprod 79.5%

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

      10. sqr-neg 79.5%

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

      11. sqrt-unprod 30.6%

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

      12. add-sqr-sqrt 76.4%

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

      13. sub-neg 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in t1 around 0 77.5%

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

   if -2.6000000000000001e-101 < u < 6.0000000000000003e-36

   1. Initial program 63.4%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t1 around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

   7. Simplified 85.0%

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

   if 6.0000000000000003e-36 < u

   1. Initial program 75.6%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around 0 78.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{-36}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{\left(-t1\right) - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.6e-101)
   (/ t1 (* (/ u v) (- t1 u)))
   (if (<= u 2.3e-48) (/ (- v) t1) (/ (* v (/ t1 u)) (- (- t1) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e-101) {
		tmp = t1 / ((u / v) * (t1 - u));
	} else if (u <= 2.3e-48) {
		tmp = -v / t1;
	} else {
		tmp = (v * (t1 / 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 <= (-2.6d-101)) then
        tmp = t1 / ((u / v) * (t1 - u))
    else if (u <= 2.3d-48) then
        tmp = -v / t1
    else
        tmp = (v * (t1 / u)) / (-t1 - u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e-101) {
		tmp = t1 / ((u / v) * (t1 - u));
	} else if (u <= 2.3e-48) {
		tmp = -v / t1;
	} else {
		tmp = (v * (t1 / u)) / (-t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.6e-101:
		tmp = t1 / ((u / v) * (t1 - u))
	elif u <= 2.3e-48:
		tmp = -v / t1
	else:
		tmp = (v * (t1 / u)) / (-t1 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.6e-101)
		tmp = Float64(t1 / Float64(Float64(u / v) * Float64(t1 - u)));
	elseif (u <= 2.3e-48)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(Float64(-t1) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.6e-101)
		tmp = t1 / ((u / v) * (t1 - u));
	elseif (u <= 2.3e-48)
		tmp = -v / t1;
	else
		tmp = (v * (t1 / u)) / (-t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.6e-101], N[(t1 / N[(N[(u / v), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.3e-48], N[((-v) / t1), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.6000000000000001e-101

   1. Initial program 75.7%

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

      1. times-frac 98.6%

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

   3. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. clear-num 98.6%

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

      3. frac-2neg 98.6%

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

      4. frac-times 85.1%

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

      5. *-un-lft-identity 85.1%

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

      6. remove-double-neg 85.1%

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

      7. distribute-neg-in 85.1%

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

      8. add-sqr-sqrt 51.1%

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

      9. sqrt-unprod 79.5%

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

      10. sqr-neg 79.5%

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

      11. sqrt-unprod 30.6%

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

      12. add-sqr-sqrt 76.4%

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

      13. sub-neg 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in t1 around 0 77.5%

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

   if -2.6000000000000001e-101 < u < 2.3000000000000001e-48

   1. Initial program 62.5%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in t1 around inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   7. Simplified 86.9%

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

   if 2.3000000000000001e-48 < u

   1. Initial program 76.0%

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

      1. associate-/r* 85.5%

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

      2. *-commutative 85.5%

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

      3. associate-/l* 96.7%

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

      4. associate-/l/ 90.8%

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

      5. +-commutative 90.8%

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

      6. remove-double-neg 90.8%

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

      7. unsub-neg 90.8%

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

      8. div-sub 90.8%

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

      9. sub-neg 90.8%

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

      10. *-inverses 90.8%

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

      11. metadata-eval 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in u around inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. distribute-neg-frac 69.8%

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

   7. Simplified 69.8%

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

      1. div-inv 69.7%

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

      2. frac-2neg 69.7%

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

      3. metadata-eval 69.7%

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

      4. distribute-rgt-neg-in 69.7%

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

      5. frac-2neg 69.7%

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

      6. remove-double-neg 69.7%

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

      7. distribute-frac-neg 69.7%

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

      8. frac-2neg 69.7%

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

   9. Applied egg-rr 69.7%

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

       1. associate-*r/ 69.8%

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

       2. *-commutative 69.8%

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

       3. times-frac 75.6%

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

       4. *-commutative 75.6%

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

       5. associate-*l/ 75.6%

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

       6. associate-/r/ 77.0%

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

       7. associate-*r* 77.0%

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

       8. associate-*r/ 77.0%

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

       9. neg-mul-1 77.0%

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

       10. +-commutative 77.0%

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

   11. Simplified 77.0%

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

       1. frac-2neg 77.0%

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

       2. +-commutative 77.0%

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

       3. div-inv 77.0%

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

       4. *-commutative 77.0%

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

       5. distribute-rgt-neg-in 77.0%

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

       6. frac-2neg 77.0%

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

       7. remove-double-neg 77.0%

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

       8. distribute-frac-neg 77.0%

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

       9. frac-2neg 77.0%

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

   13. Applied egg-rr 77.0%

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

       1. associate-*r/ 77.0%

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

       2. *-rgt-identity 77.0%

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

       3. associate-*r/ 73.7%

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

       4. associate-/l* 78.2%

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

       5. associate-/r/ 77.0%

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

       6. distribute-neg-in 77.0%

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

       7. unsub-neg 77.0%

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

   15. Simplified 77.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.6e-101)
   (/ t1 (* (/ u v) (- t1 u)))
   (if (<= u 2.6e-17) (/ (- v) t1) (/ (* (- t1) (/ v u)) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e-101) {
		tmp = t1 / ((u / v) * (t1 - u));
	} else if (u <= 2.6e-17) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 * (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 <= (-2.6d-101)) then
        tmp = t1 / ((u / v) * (t1 - u))
    else if (u <= 2.6d-17) then
        tmp = -v / t1
    else
        tmp = (-t1 * (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 <= -2.6e-101) {
		tmp = t1 / ((u / v) * (t1 - u));
	} else if (u <= 2.6e-17) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 * (v / u)) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.6e-101:
		tmp = t1 / ((u / v) * (t1 - u))
	elif u <= 2.6e-17:
		tmp = -v / t1
	else:
		tmp = (-t1 * (v / u)) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.6e-101)
		tmp = Float64(t1 / Float64(Float64(u / v) * Float64(t1 - u)));
	elseif (u <= 2.6e-17)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(-t1) * Float64(v / u)) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.6e-101)
		tmp = t1 / ((u / v) * (t1 - u));
	elseif (u <= 2.6e-17)
		tmp = -v / t1;
	else
		tmp = (-t1 * (v / u)) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.6e-101], N[(t1 / N[(N[(u / v), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.6e-17], N[((-v) / t1), $MachinePrecision], N[(N[((-t1) * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.6000000000000001e-101

   1. Initial program 75.7%

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

      1. times-frac 98.6%

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

   3. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. clear-num 98.6%

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

      3. frac-2neg 98.6%

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

      4. frac-times 85.1%

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

      5. *-un-lft-identity 85.1%

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

      6. remove-double-neg 85.1%

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

      7. distribute-neg-in 85.1%

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

      8. add-sqr-sqrt 51.1%

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

      9. sqrt-unprod 79.5%

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

      10. sqr-neg 79.5%

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

      11. sqrt-unprod 30.6%

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

      12. add-sqr-sqrt 76.4%

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

      13. sub-neg 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in t1 around 0 77.5%

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

   if -2.6000000000000001e-101 < u < 2.60000000000000003e-17

   1. Initial program 63.0%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t1 around inf 83.8%

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

      1. associate-*r/ 83.8%

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

      2. neg-mul-1 83.8%

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

   7. Simplified 83.8%

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

   if 2.60000000000000003e-17 < u

   1. Initial program 77.3%

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

      1. associate-/r* 84.8%

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

      2. *-commutative 84.8%

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

      3. associate-/l* 96.2%

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

      4. associate-/l/ 89.2%

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

      5. +-commutative 89.2%

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

      6. remove-double-neg 89.2%

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

      7. unsub-neg 89.2%

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

      8. div-sub 89.2%

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

      9. sub-neg 89.2%

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

      10. *-inverses 89.2%

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

      11. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in u around inf 72.1%

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

      1. mul-1-neg 72.1%

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

      2. distribute-neg-frac 72.1%

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

   7. Simplified 72.1%

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

      1. div-inv 72.0%

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

      2. frac-2neg 72.0%

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

      3. metadata-eval 72.0%

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

      4. distribute-rgt-neg-in 72.0%

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

      5. frac-2neg 72.0%

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

      6. remove-double-neg 72.0%

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

      7. distribute-frac-neg 72.0%

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

      8. frac-2neg 72.0%

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

   9. Applied egg-rr 72.0%

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

       1. associate-*r/ 72.1%

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

       2. *-commutative 72.1%

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

       3. times-frac 79.0%

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

       4. *-commutative 79.0%

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

       5. associate-*l/ 79.0%

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

       6. associate-/r/ 82.5%

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

       7. associate-*r* 82.5%

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

       8. associate-*r/ 82.5%

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

       9. neg-mul-1 82.5%

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

       10. +-commutative 82.5%

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

   11. Simplified 82.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{-u}{t1}}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.6e-101)
   (/ (/ v u) (/ (- u) t1))
   (if (<= u 2.3e-48) (/ (- v) t1) (* (/ v u) (/ t1 (- t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e-101) {
		tmp = (v / u) / (-u / t1);
	} else if (u <= 2.3e-48) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (t1 / (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 <= (-2.6d-101)) then
        tmp = (v / u) / (-u / t1)
    else if (u <= 2.3d-48) then
        tmp = -v / t1
    else
        tmp = (v / u) * (t1 / (t1 - u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e-101) {
		tmp = (v / u) / (-u / t1);
	} else if (u <= 2.3e-48) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (t1 / (t1 - u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.6e-101:
		tmp = (v / u) / (-u / t1)
	elif u <= 2.3e-48:
		tmp = -v / t1
	else:
		tmp = (v / u) * (t1 / (t1 - u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.6e-101)
		tmp = Float64(Float64(v / u) / Float64(Float64(-u) / t1));
	elseif (u <= 2.3e-48)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(t1 - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.6e-101)
		tmp = (v / u) / (-u / t1);
	elseif (u <= 2.3e-48)
		tmp = -v / t1;
	else
		tmp = (v / u) * (t1 / (t1 - u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.6e-101], N[(N[(v / u), $MachinePrecision] / N[((-u) / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.3e-48], N[((-v) / t1), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.6000000000000001e-101

   1. Initial program 75.7%

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

      1. associate-/r* 87.2%

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

      2. *-commutative 87.2%

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

      3. associate-/l* 98.7%

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

      4. associate-/l/ 96.6%

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

      5. +-commutative 96.6%

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

      6. remove-double-neg 96.6%

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

      7. unsub-neg 96.6%

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

      8. div-sub 96.6%

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

      9. sub-neg 96.6%

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

      10. *-inverses 96.6%

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

      11. metadata-eval 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in v around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. mul-1-neg 96.6%

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

      3. distribute-neg-in 96.6%

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

      4. associate-/r* 98.7%

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

      5. +-commutative 98.7%

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

      6. distribute-neg-in 98.7%

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

      7. metadata-eval 98.7%

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

      8. sub-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in t1 around 0 76.2%

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

   9. Taylor expanded in u around inf 77.2%

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

       1. associate-*r/ 77.2%

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

       2. mul-1-neg 77.2%

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

   11. Simplified 77.2%

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

   if -2.6000000000000001e-101 < u < 2.3000000000000001e-48

   1. Initial program 62.5%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in t1 around inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   7. Simplified 86.9%

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

   if 2.3000000000000001e-48 < u

   1. Initial program 76.0%

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

      1. times-frac 96.5%

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

   3. Simplified 96.5%

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

      1. *-commutative 96.5%

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

      2. clear-num 95.3%

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

      3. frac-2neg 95.3%

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

      4. frac-times 86.0%

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

      5. *-un-lft-identity 86.0%

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

      6. remove-double-neg 86.0%

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

      7. distribute-neg-in 86.0%

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

      8. add-sqr-sqrt 45.3%

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

      9. sqrt-unprod 78.1%

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

      10. sqr-neg 78.1%

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

      11. sqrt-unprod 34.2%

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

      12. add-sqr-sqrt 73.6%

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

      13. sub-neg 73.6%

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

   6. Applied egg-rr 73.6%

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

   7. Taylor expanded in t1 around 0 73.6%

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

   8. Taylor expanded in v around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. times-frac 75.4%

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

   10. Simplified 75.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{-u}{t1}}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.7 \cdot 10^{-102}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.7e-102)
   (/ t1 (* (/ u v) (- t1 u)))
   (if (<= u 2.3e-48) (/ (- v) t1) (* (/ v u) (/ t1 (- t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.7e-102) {
		tmp = t1 / ((u / v) * (t1 - u));
	} else if (u <= 2.3e-48) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (t1 / (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 <= (-4.7d-102)) then
        tmp = t1 / ((u / v) * (t1 - u))
    else if (u <= 2.3d-48) then
        tmp = -v / t1
    else
        tmp = (v / u) * (t1 / (t1 - u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.7e-102) {
		tmp = t1 / ((u / v) * (t1 - u));
	} else if (u <= 2.3e-48) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (t1 / (t1 - u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.7e-102:
		tmp = t1 / ((u / v) * (t1 - u))
	elif u <= 2.3e-48:
		tmp = -v / t1
	else:
		tmp = (v / u) * (t1 / (t1 - u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.7e-102)
		tmp = Float64(t1 / Float64(Float64(u / v) * Float64(t1 - u)));
	elseif (u <= 2.3e-48)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(t1 - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.7e-102)
		tmp = t1 / ((u / v) * (t1 - u));
	elseif (u <= 2.3e-48)
		tmp = -v / t1;
	else
		tmp = (v / u) * (t1 / (t1 - u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.7e-102], N[(t1 / N[(N[(u / v), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.3e-48], N[((-v) / t1), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.7000000000000003e-102

   1. Initial program 75.7%

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

      1. times-frac 98.6%

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

   3. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. clear-num 98.6%

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

      3. frac-2neg 98.6%

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

      4. frac-times 85.1%

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

      5. *-un-lft-identity 85.1%

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

      6. remove-double-neg 85.1%

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

      7. distribute-neg-in 85.1%

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

      8. add-sqr-sqrt 51.1%

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

      9. sqrt-unprod 79.5%

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

      10. sqr-neg 79.5%

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

      11. sqrt-unprod 30.6%

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

      12. add-sqr-sqrt 76.4%

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

      13. sub-neg 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in t1 around 0 77.5%

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

   if -4.7000000000000003e-102 < u < 2.3000000000000001e-48

   1. Initial program 62.5%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in t1 around inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   7. Simplified 86.9%

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

   if 2.3000000000000001e-48 < u

   1. Initial program 76.0%

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

      1. times-frac 96.5%

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

   3. Simplified 96.5%

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

      1. *-commutative 96.5%

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

      2. clear-num 95.3%

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

      3. frac-2neg 95.3%

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

      4. frac-times 86.0%

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

      5. *-un-lft-identity 86.0%

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

      6. remove-double-neg 86.0%

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

      7. distribute-neg-in 86.0%

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

      8. add-sqr-sqrt 45.3%

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

      9. sqrt-unprod 78.1%

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

      10. sqr-neg 78.1%

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

      11. sqrt-unprod 34.2%

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

      12. add-sqr-sqrt 73.6%

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

      13. sub-neg 73.6%

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

   6. Applied egg-rr 73.6%

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

   7. Taylor expanded in t1 around 0 73.6%

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

   8. Taylor expanded in v around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. times-frac 75.4%

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

   10. Simplified 75.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.7 \cdot 10^{-102}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{-48}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-30} \lor \neg \left(t1 \leq 1.85 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{-u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.2e-30) (not (<= t1 1.85e-150)))
   (/ v (- (* u -2.0) t1))
   (/ (/ v u) (/ (- u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.2e-30) || !(t1 <= 1.85e-150)) {
		tmp = v / ((u * -2.0) - 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 ((t1 <= (-1.2d-30)) .or. (.not. (t1 <= 1.85d-150))) then
        tmp = v / ((u * (-2.0d0)) - 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 ((t1 <= -1.2e-30) || !(t1 <= 1.85e-150)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v / u) / (-u / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.2e-30) or not (t1 <= 1.85e-150):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (v / u) / (-u / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.2e-30) || !(t1 <= 1.85e-150))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(v / u) / Float64(Float64(-u) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.2e-30) || ~((t1 <= 1.85e-150)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (v / u) / (-u / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.2e-30], N[Not[LessEqual[t1, 1.85e-150]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] / N[((-u) / t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.19999999999999992e-30 or 1.85e-150 < t1

   1. Initial program 67.5%

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

      1. associate-/r* 81.0%

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

      2. *-commutative 81.0%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 96.8%

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

      5. +-commutative 96.8%

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

      6. remove-double-neg 96.8%

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

      7. unsub-neg 96.8%

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

      8. div-sub 96.8%

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

      9. sub-neg 96.8%

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

      10. *-inverses 96.8%

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

      11. metadata-eval 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in t1 around inf 80.6%

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

      1. mul-1-neg 80.6%

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

      2. unsub-neg 80.6%

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

      3. *-commutative 80.6%

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

   7. Simplified 80.6%

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

   if -1.19999999999999992e-30 < t1 < 1.85e-150

   1. Initial program 74.0%

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

      1. associate-/r* 83.7%

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

      2. *-commutative 83.7%

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

      3. associate-/l* 95.5%

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

      4. associate-/l/ 96.3%

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

      5. +-commutative 96.3%

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

      6. remove-double-neg 96.3%

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

      7. unsub-neg 96.3%

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

      8. div-sub 96.3%

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

      9. sub-neg 96.3%

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

      10. *-inverses 96.3%

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

      11. metadata-eval 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in v around 0 96.3%

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

      1. sub-neg 96.3%

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

      2. mul-1-neg 96.3%

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

      3. distribute-neg-in 96.3%

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

      4. associate-/r* 94.5%

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

      5. +-commutative 94.5%

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

      6. distribute-neg-in 94.5%

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

      7. metadata-eval 94.5%

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

      8. sub-neg 94.5%

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

   7. Simplified 94.5%

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

   8. Taylor expanded in t1 around 0 76.2%

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

   9. Taylor expanded in u around inf 81.1%

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

       1. associate-*r/ 81.1%

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

       2. mul-1-neg 81.1%

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

   11. Simplified 81.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-30} \lor \neg \left(t1 \leq 1.85 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{-u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+118} \lor \neg \left(u \leq 7 \cdot 10^{+196}\right):\\ \;\;\;\;\frac{t1}{\frac{t1}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.8e+118) (not (<= u 7e+196)))
   (/ t1 (/ t1 (/ v u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.8e+118) || !(u <= 7e+196)) {
		tmp = t1 / (t1 / (v / u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.8d+118)) .or. (.not. (u <= 7d+196))) then
        tmp = t1 / (t1 / (v / u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.8e+118) || !(u <= 7e+196)) {
		tmp = t1 / (t1 / (v / u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.8e+118) or not (u <= 7e+196):
		tmp = t1 / (t1 / (v / u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.8e+118) || !(u <= 7e+196))
		tmp = Float64(t1 / Float64(t1 / Float64(v / u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.8e+118) || ~((u <= 7e+196)))
		tmp = t1 / (t1 / (v / u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.8e+118], N[Not[LessEqual[u, 7e+196]], $MachinePrecision]], N[(t1 / N[(t1 / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.79999999999999986e118 or 6.9999999999999997e196 < u

   1. Initial program 79.8%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.9%

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

      3. frac-2neg 99.9%

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

      4. frac-times 86.8%

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

      5. *-un-lft-identity 86.8%

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

      6. remove-double-neg 86.8%

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

      7. distribute-neg-in 86.8%

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

      8. add-sqr-sqrt 42.4%

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

      9. sqrt-unprod 85.1%

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

      10. sqr-neg 85.1%

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

      11. sqrt-unprod 44.4%

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

      12. add-sqr-sqrt 86.8%

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

      13. sub-neg 86.8%

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

   6. Applied egg-rr 86.8%

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

   7. Taylor expanded in t1 around 0 86.8%

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

   8. Taylor expanded in u around 0 55.2%

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

      1. associate-/l* 55.1%

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

   10. Simplified 55.1%

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

   if -2.79999999999999986e118 < u < 6.9999999999999997e196

   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 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t1 around inf 62.3%

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

      1. associate-*r/ 62.3%

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

      2. neg-mul-1 62.3%

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

   7. Simplified 62.3%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+118} \lor \neg \left(u \leq 7 \cdot 10^{+196}\right):\\ \;\;\;\;\frac{t1}{\frac{t1}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+117} \lor \neg \left(u \leq 6.2 \cdot 10^{+196}\right):\\ \;\;\;\;\frac{t1}{\frac{t1 \cdot u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.3e+117) (not (<= u 6.2e+196)))
   (/ t1 (/ (* t1 u) v))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e+117) || !(u <= 6.2e+196)) {
		tmp = t1 / ((t1 * 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 <= (-2.3d+117)) .or. (.not. (u <= 6.2d+196))) then
        tmp = t1 / ((t1 * 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 <= -2.3e+117) || !(u <= 6.2e+196)) {
		tmp = t1 / ((t1 * u) / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.3e+117) or not (u <= 6.2e+196):
		tmp = t1 / ((t1 * u) / v)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.3e+117) || !(u <= 6.2e+196))
		tmp = Float64(t1 / Float64(Float64(t1 * u) / v));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.3e+117) || ~((u <= 6.2e+196)))
		tmp = t1 / ((t1 * u) / v);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.3e+117], N[Not[LessEqual[u, 6.2e+196]], $MachinePrecision]], N[(t1 / N[(N[(t1 * u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.29999999999999988e117 or 6.2000000000000002e196 < u

   1. Initial program 79.8%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.9%

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

      3. frac-2neg 99.9%

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

      4. frac-times 86.8%

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

      5. *-un-lft-identity 86.8%

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

      6. remove-double-neg 86.8%

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

      7. distribute-neg-in 86.8%

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

      8. add-sqr-sqrt 42.4%

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

      9. sqrt-unprod 85.1%

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

      10. sqr-neg 85.1%

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

      11. sqrt-unprod 44.4%

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

      12. add-sqr-sqrt 86.8%

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

      13. sub-neg 86.8%

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

   6. Applied egg-rr 86.8%

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

   7. Taylor expanded in t1 around 0 86.8%

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

   8. Taylor expanded in u around 0 55.2%

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

   if -2.29999999999999988e117 < u < 6.2000000000000002e196

   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 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t1 around inf 62.3%

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

      1. associate-*r/ 62.3%

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

      2. neg-mul-1 62.3%

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

   7. Simplified 62.3%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+117} \lor \neg \left(u \leq 6.2 \cdot 10^{+196}\right):\\ \;\;\;\;\frac{t1}{\frac{t1 \cdot u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.6 \cdot 10^{+120} \lor \neg \left(u \leq 2.1 \cdot 10^{+221}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -8.6e+120) (not (<= u 2.1e+221))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.6e+120) || !(u <= 2.1e+221)) {
		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 <= (-8.6d+120)) .or. (.not. (u <= 2.1d+221))) 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 <= -8.6e+120) || !(u <= 2.1e+221)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -8.6e+120) or not (u <= 2.1e+221):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -8.6e+120) || !(u <= 2.1e+221))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -8.6e+120) || ~((u <= 2.1e+221)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -8.6e+120], N[Not[LessEqual[u, 2.1e+221]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -8.6000000000000003e120 or 2.10000000000000002e221 < u

   1. Initial program 79.3%

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

      1. associate-/r* 95.9%

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

      2. *-commutative 95.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 88.6%

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

      5. +-commutative 88.6%

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

      6. remove-double-neg 88.6%

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

      7. unsub-neg 88.6%

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

      8. div-sub 88.6%

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

      9. sub-neg 88.6%

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

      10. *-inverses 88.6%

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

      11. metadata-eval 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in u around inf 80.9%

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

      1. mul-1-neg 80.9%

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

      2. distribute-neg-frac 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in t1 around inf 47.7%

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

      1. associate-*r/ 47.7%

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

      2. neg-mul-1 47.7%

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

   10. Simplified 47.7%

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

   if -8.6000000000000003e120 < u < 2.10000000000000002e221

   1. Initial program 68.1%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t1 around inf 61.9%

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

      1. associate-*r/ 61.9%

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

      2. neg-mul-1 61.9%

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

   7. Simplified 61.9%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.6 \cdot 10^{+120} \lor \neg \left(u \leq 2.1 \cdot 10^{+221}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. associate-/r* 82.1%

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

   2. *-commutative 82.1%

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

   3. associate-/l* 98.1%

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

   4. associate-/l/ 96.6%

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

   5. +-commutative 96.6%

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

   6. remove-double-neg 96.6%

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

   7. unsub-neg 96.6%

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

   8. div-sub 96.6%

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

   9. sub-neg 96.6%

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

   10. *-inverses 96.6%

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

   11. metadata-eval 96.6%

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

3. Simplified 96.6%

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

5. Taylor expanded in u around 0 96.6%

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

   1. sub-neg 96.6%

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

   2. mul-1-neg 96.6%

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

   3. distribute-neg-in 96.6%

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

   4. +-commutative 96.6%

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

   5. distribute-neg-in 96.6%

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

   6. metadata-eval 96.6%

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

   7. sub-neg 96.6%

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 13: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. associate-/r* 82.1%

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

   2. *-commutative 82.1%

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

   3. associate-/l* 98.1%

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

   4. associate-/l/ 96.6%

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

   5. +-commutative 96.6%

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

   6. remove-double-neg 96.6%

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

   7. unsub-neg 96.6%

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

   8. div-sub 96.6%

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

   9. sub-neg 96.6%

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

   10. *-inverses 96.6%

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

   11. metadata-eval 96.6%

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

3. Simplified 96.6%

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

5. Taylor expanded in v around 0 96.6%

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

   1. sub-neg 96.6%

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

   2. mul-1-neg 96.6%

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

   3. distribute-neg-in 96.6%

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

   4. associate-/r* 97.6%

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

   5. +-commutative 97.6%

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

   6. distribute-neg-in 97.6%

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

   7. metadata-eval 97.6%

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

   8. sub-neg 97.6%

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

7. Simplified 97.6%

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

8. Final simplification 97.6%

   \[\leadsto \frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}} \]
9. Add Preprocessing

Alternative 14: 62.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / ((u * (-2.0d0)) - t1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. associate-/r* 82.1%

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

   2. *-commutative 82.1%

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

   3. associate-/l* 98.1%

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

   4. associate-/l/ 96.6%

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

   5. +-commutative 96.6%

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

   6. remove-double-neg 96.6%

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

   7. unsub-neg 96.6%

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

   8. div-sub 96.6%

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

   9. sub-neg 96.6%

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

   10. *-inverses 96.6%

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

   11. metadata-eval 96.6%

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

3. Simplified 96.6%

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

5. Taylor expanded in t1 around inf 61.4%

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

   1. mul-1-neg 61.4%

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

   2. unsub-neg 61.4%

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

   3. *-commutative 61.4%

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

7. Simplified 61.4%

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

8. Final simplification 61.4%

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

Alternative 15: 54.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. times-frac 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in t1 around inf 53.8%

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

   1. associate-*r/ 53.8%

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

   2. neg-mul-1 53.8%

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

7. Simplified 53.8%

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

8. Final simplification 53.8%

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

Reproduce

herbie shell --seed 2024018 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))