Rosa's DopplerBench

Percentage Accurate: 71.9% → 98.1%

Time: 10.0s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. times-frac 97.3%

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

   2. distribute-frac-neg 97.3%

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

   3. distribute-neg-frac2 97.3%

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

   4. +-commutative 97.3%

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

   5. distribute-neg-in 97.3%

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

   6. unsub-neg 97.3%

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

3. Simplified 97.3%

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

   1. frac-2neg 97.3%

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

   2. frac-2neg 97.3%

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

   3. frac-times 68.9%

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

   4. sub-neg 68.9%

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

   5. distribute-neg-in 68.9%

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

   6. +-commutative 68.9%

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

   7. remove-double-neg 68.9%

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

   8. frac-times 97.3%

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

   9. associate-*r/ 98.2%

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

   10. add-sqr-sqrt 50.3%

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

   11. sqrt-unprod 43.5%

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

   12. sqr-neg 43.5%

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

   13. sqrt-unprod 15.5%

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

   14. add-sqr-sqrt 33.0%

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

   15. add-sqr-sqrt 13.9%

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

   16. sqrt-unprod 53.2%

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

6. Applied egg-rr 98.2%

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

7. Final simplification 98.2%

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

Alternative 2: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+136} \lor \neg \left(t1 \leq 1.85 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.5e+136) (not (<= t1 1.85e+118)))
   (/ v (- u t1))
   (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e+136) || !(t1 <= 1.85e+118)) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4.5d+136)) .or. (.not. (t1 <= 1.85d+118))) then
        tmp = v / (u - t1)
    else
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e+136) || !(t1 <= 1.85e+118)) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.5e+136) or not (t1 <= 1.85e+118):
		tmp = v / (u - t1)
	else:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.5e+136) || !(t1 <= 1.85e+118))
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.5e+136) || ~((t1 <= 1.85e+118)))
		tmp = v / (u - t1);
	else
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.5e+136], N[Not[LessEqual[t1, 1.85e+118]], $MachinePrecision]], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.4999999999999999e136 or 1.84999999999999993e118 < t1

   1. Initial program 42.9%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 90.8%

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

      1. add-sqr-sqrt 34.8%

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

      2. add-sqr-sqrt 18.6%

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

      3. difference-of-squares 18.6%

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

      4. add-sqr-sqrt 18.6%

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

      5. sqrt-unprod 17.4%

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

      6. sqr-neg 17.4%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 26.8%

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

      11. sqr-neg 26.8%

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

      12. sqrt-unprod 30.6%

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

      13. add-sqr-sqrt 30.6%

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

   7. Applied egg-rr 30.6%

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

      1. difference-of-squares 30.6%

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

      2. rem-square-sqrt 49.3%

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

      3. rem-square-sqrt 91.0%

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

   9. Simplified 91.0%

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

   10. Taylor expanded in v around 0 91.0%

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

   if -4.4999999999999999e136 < t1 < 1.84999999999999993e118

   1. Initial program 79.7%

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

      1. associate-/l* 78.1%

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

      2. distribute-lft-neg-out 78.1%

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

      3. distribute-rgt-neg-in 78.1%

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

      4. associate-/r* 89.1%

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

      5. distribute-neg-frac2 89.1%

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

   3. Simplified 89.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+136} \lor \neg \left(t1 \leq 1.85 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{-51} \lor \neg \left(t1 \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.5e-51) (not (<= t1 1.15e-15)))
   (/ v (- u t1))
   (/ (* v (/ t1 (- u))) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.5e-51) || !(t1 <= 1.15e-15)) {
		tmp = v / (u - 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 ((t1 <= (-9.5d-51)) .or. (.not. (t1 <= 1.15d-15))) then
        tmp = v / (u - 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 ((t1 <= -9.5e-51) || !(t1 <= 1.15e-15)) {
		tmp = v / (u - t1);
	} else {
		tmp = (v * (t1 / -u)) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9.5e-51) or not (t1 <= 1.15e-15):
		tmp = v / (u - t1)
	else:
		tmp = (v * (t1 / -u)) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9.5e-51) || !(t1 <= 1.15e-15))
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(Float64(v * Float64(t1 / Float64(-u))) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -9.5e-51) || ~((t1 <= 1.15e-15)))
		tmp = v / (u - t1);
	else
		tmp = (v * (t1 / -u)) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9.5e-51], N[Not[LessEqual[t1, 1.15e-15]], $MachinePrecision]], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.4999999999999998e-51 or 1.14999999999999995e-15 < t1

   1. Initial program 56.3%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 77.9%

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

      1. add-sqr-sqrt 31.7%

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

      2. add-sqr-sqrt 16.7%

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

      3. difference-of-squares 16.7%

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

      4. add-sqr-sqrt 16.7%

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

      5. sqrt-unprod 16.8%

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

      6. sqr-neg 16.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 18.2%

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

      11. sqr-neg 18.2%

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

      12. sqrt-unprod 20.1%

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

      13. add-sqr-sqrt 20.1%

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

   7. Applied egg-rr 20.1%

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

      1. difference-of-squares 20.1%

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

      2. rem-square-sqrt 37.0%

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

      3. rem-square-sqrt 78.2%

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

   9. Simplified 78.2%

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

   10. Taylor expanded in v around 0 78.2%

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

   if -9.4999999999999998e-51 < t1 < 1.14999999999999995e-15

   1. Initial program 85.0%

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

      1. times-frac 93.9%

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

      2. distribute-frac-neg 93.9%

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

      3. distribute-neg-frac2 93.9%

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

      4. +-commutative 93.9%

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

      5. distribute-neg-in 93.9%

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

      6. unsub-neg 93.9%

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

   3. Simplified 93.9%

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

      1. frac-2neg 93.9%

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

      2. frac-2neg 93.9%

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

      3. frac-times 85.0%

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

      4. sub-neg 85.0%

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

      5. distribute-neg-in 85.0%

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

      6. +-commutative 85.0%

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

      7. remove-double-neg 85.0%

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

      8. frac-times 93.9%

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

      9. associate-*r/ 96.0%

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

      10. add-sqr-sqrt 45.1%

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

      11. sqrt-unprod 49.6%

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

      12. sqr-neg 49.6%

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

      13. sqrt-unprod 18.9%

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

      14. add-sqr-sqrt 41.1%

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

      15. add-sqr-sqrt 15.8%

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

      16. sqrt-unprod 67.2%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in t1 around 0 80.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{-51} \lor \neg \left(t1 \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.6 \cdot 10^{-44} \lor \neg \left(t1 \leq 1.42 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.6e-44) (not (<= t1 1.42e-14)))
   (/ v (- u t1))
   (* t1 (/ (/ v u) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.6e-44) || !(t1 <= 1.42e-14)) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3.6d-44)) .or. (.not. (t1 <= 1.42d-14))) then
        tmp = v / (u - t1)
    else
        tmp = t1 * ((v / u) / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.6e-44) || !(t1 <= 1.42e-14)) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.6e-44) or not (t1 <= 1.42e-14):
		tmp = v / (u - t1)
	else:
		tmp = t1 * ((v / u) / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.6e-44) || !(t1 <= 1.42e-14))
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.6e-44) || ~((t1 <= 1.42e-14)))
		tmp = v / (u - t1);
	else
		tmp = t1 * ((v / u) / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.6e-44], N[Not[LessEqual[t1, 1.42e-14]], $MachinePrecision]], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.5999999999999999e-44 or 1.42000000000000004e-14 < t1

   1. Initial program 55.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 78.3%

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

      1. add-sqr-sqrt 32.2%

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

      2. add-sqr-sqrt 16.9%

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

      3. difference-of-squares 16.9%

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

      4. add-sqr-sqrt 16.9%

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

      5. sqrt-unprod 17.0%

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

      6. sqr-neg 17.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 18.5%

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

      11. sqr-neg 18.5%

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

      12. sqrt-unprod 20.4%

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

      13. add-sqr-sqrt 20.4%

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

   7. Applied egg-rr 20.4%

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

      1. difference-of-squares 20.4%

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

      2. rem-square-sqrt 37.5%

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

      3. rem-square-sqrt 78.5%

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

   9. Simplified 78.5%

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

   10. Taylor expanded in v around 0 78.5%

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

   if -3.5999999999999999e-44 < t1 < 1.42000000000000004e-14

   1. Initial program 85.3%

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

      1. associate-/l* 81.3%

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

      2. distribute-lft-neg-out 81.3%

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

      3. distribute-rgt-neg-in 81.3%

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

      4. associate-/r* 90.7%

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

      5. distribute-neg-frac2 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t1 around 0 78.1%

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

   6. Taylor expanded in t1 around 0 80.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.6 \cdot 10^{-44} \lor \neg \left(t1 \leq 1.42 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+112} \lor \neg \left(u \leq 4 \cdot 10^{+131}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.4e+112) (not (<= u 4e+131)))
   (* v (/ (/ t1 u) u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+112) || !(u <= 4e+131)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.4d+112)) .or. (.not. (u <= 4d+131))) then
        tmp = v * ((t1 / u) / u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+112) || !(u <= 4e+131)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.4e+112) or not (u <= 4e+131):
		tmp = v * ((t1 / u) / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.4e+112) || !(u <= 4e+131))
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.4e+112) || ~((u <= 4e+131)))
		tmp = v * ((t1 / u) / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.4e+112], N[Not[LessEqual[u, 4e+131]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.39999999999999993e112 or 3.9999999999999996e131 < u

   1. Initial program 65.4%

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

      1. associate-*l/ 63.8%

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

      2. *-commutative 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in t1 around 0 63.2%

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

   6. Taylor expanded in t1 around 0 63.2%

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

      1. associate-/r* 71.4%

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

      2. div-inv 71.4%

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

      3. add-sqr-sqrt 41.6%

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

      4. sqrt-unprod 53.0%

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

      5. sqr-neg 53.0%

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

      6. sqrt-unprod 21.2%

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

      7. add-sqr-sqrt 56.4%

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

   8. Applied egg-rr 56.4%

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

      1. associate-*r/ 56.4%

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

      2. *-rgt-identity 56.4%

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

   10. Simplified 56.4%

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

   if -3.39999999999999993e112 < u < 3.9999999999999996e131

   1. Initial program 70.9%

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

      1. times-frac 96.9%

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

      2. distribute-frac-neg 96.9%

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

      3. distribute-neg-frac2 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. unsub-neg 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in t1 around inf 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   7. Simplified 70.2%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+112} \lor \neg \left(u \leq 4 \cdot 10^{+131}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 23.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.9 \cdot 10^{+49} \lor \neg \left(t1 \leq 2.7 \cdot 10^{+125}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.9e+49) (not (<= t1 2.7e+125))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.9e+49) || !(t1 <= 2.7e+125)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3.9d+49)) .or. (.not. (t1 <= 2.7d+125))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.9e+49) || !(t1 <= 2.7e+125)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.9e+49) or not (t1 <= 2.7e+125):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.9e+49) || !(t1 <= 2.7e+125))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.9e+49) || ~((t1 <= 2.7e+125)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.9e+49], N[Not[LessEqual[t1, 2.7e+125]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.9000000000000001e49 or 2.6999999999999999e125 < t1

   1. Initial program 43.9%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 82.0%

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

      1. associate-*r/ 82.0%

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

      2. neg-mul-1 82.0%

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

   7. Simplified 82.0%

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

      1. distribute-frac-neg 82.0%

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

      2. div-inv 81.9%

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

      3. distribute-rgt-neg-in 81.9%

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

      4. frac-2neg 81.9%

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

      5. metadata-eval 81.9%

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

      6. add-sqr-sqrt 46.5%

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

      7. sqrt-unprod 43.4%

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

      8. sqr-neg 43.4%

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

      9. sqrt-unprod 12.7%

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

      10. add-sqr-sqrt 26.1%

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

   9. Applied egg-rr 26.1%

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

       1. distribute-rgt-neg-out 26.1%

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

       2. *-commutative 26.1%

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

       3. associate-*l/ 26.1%

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

       4. mul-1-neg 26.1%

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

       5. distribute-neg-frac 26.1%

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

       6. remove-double-neg 26.1%

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

   11. Simplified 26.1%

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

   if -3.9000000000000001e49 < t1 < 2.6999999999999999e125

   1. Initial program 83.6%

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

      1. times-frac 95.7%

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

      2. distribute-frac-neg 95.7%

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

      3. distribute-neg-frac2 95.7%

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

      4. +-commutative 95.7%

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

      5. distribute-neg-in 95.7%

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

      6. unsub-neg 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in t1 around inf 45.8%

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

      1. add-sqr-sqrt 22.8%

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

      2. add-sqr-sqrt 13.9%

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

      3. difference-of-squares 13.9%

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

      4. add-sqr-sqrt 13.9%

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

      5. sqrt-unprod 14.5%

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

      6. sqr-neg 14.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 12.8%

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

      11. sqr-neg 12.8%

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

      12. sqrt-unprod 10.9%

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

      13. add-sqr-sqrt 10.9%

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

   7. Applied egg-rr 10.9%

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

      1. difference-of-squares 10.9%

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

      2. rem-square-sqrt 25.1%

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

      3. rem-square-sqrt 45.2%

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

   9. Simplified 45.2%

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

   10. Taylor expanded in t1 around 0 15.1%

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

4. Final simplification 19.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.9 \cdot 10^{+49} \lor \neg \left(t1 \leq 2.7 \cdot 10^{+125}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. times-frac 97.3%

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

   2. distribute-frac-neg 97.3%

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

   3. distribute-neg-frac2 97.3%

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

   4. +-commutative 97.3%

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

   5. distribute-neg-in 97.3%

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

   6. unsub-neg 97.3%

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

3. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 8: 56.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 2.7 \cdot 10^{+195}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 2.7e+195) (/ v (- t1)) (/ v (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 2.7e+195) {
		tmp = v / -t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 2.7d+195) then
        tmp = v / -t1
    else
        tmp = v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 2.7e+195) {
		tmp = v / -t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 2.7e+195:
		tmp = v / -t1
	else:
		tmp = v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 2.7e+195)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 2.7e+195)
		tmp = v / -t1;
	else
		tmp = v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 2.7e+195], N[(v / (-t1)), $MachinePrecision], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 2.7000000000000002e195

   1. Initial program 68.2%

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

      1. times-frac 96.9%

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

      2. distribute-frac-neg 96.9%

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

      3. distribute-neg-frac2 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. unsub-neg 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in t1 around inf 55.4%

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

      1. associate-*r/ 55.4%

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

      2. neg-mul-1 55.4%

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

   7. Simplified 55.4%

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

   if 2.7000000000000002e195 < u

   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. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 45.9%

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

      1. add-sqr-sqrt 39.7%

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

      2. sqrt-unprod 57.6%

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

      3. mul-1-neg 57.6%

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

      4. mul-1-neg 57.6%

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

      5. sqr-neg 57.6%

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

      6. sqrt-unprod 37.8%

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

      7. add-sqr-sqrt 39.5%

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

   7. Applied egg-rr 39.5%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 2.7 \cdot 10^{+195}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 3 \cdot 10^{+195}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 3e+195) (/ v (- t1)) (/ v (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3e+195) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 3d+195) then
        tmp = v / -t1
    else
        tmp = v / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3e+195) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 3e+195:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 3e+195)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 3e+195)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 3e+195], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 3.0000000000000001e195

   1. Initial program 68.2%

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

      1. times-frac 96.9%

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

      2. distribute-frac-neg 96.9%

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

      3. distribute-neg-frac2 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. unsub-neg 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in t1 around inf 55.4%

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

      1. associate-*r/ 55.4%

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

      2. neg-mul-1 55.4%

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

   7. Simplified 55.4%

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

   if 3.0000000000000001e195 < u

   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. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 45.9%

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

   6. Taylor expanded in t1 around 0 39.4%

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

      1. associate-*r/ 39.4%

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

      2. mul-1-neg 39.4%

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

   8. Simplified 39.4%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 3 \cdot 10^{+195}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 2.8 \cdot 10^{+195}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= u 2.8e+195) (/ v (- t1)) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 2.8e+195) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 2.8d+195) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 2.8e+195) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 2.8e+195:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 2.8e+195)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 2.8e+195)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 2.8e+195], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 2.7999999999999998e195

   1. Initial program 68.2%

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

      1. times-frac 96.9%

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

      2. distribute-frac-neg 96.9%

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

      3. distribute-neg-frac2 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. unsub-neg 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in t1 around inf 55.4%

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

      1. associate-*r/ 55.4%

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

      2. neg-mul-1 55.4%

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

   7. Simplified 55.4%

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

   if 2.7999999999999998e195 < u

   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. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 56.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. difference-of-squares 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. sqr-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 21.0%

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

      11. sqr-neg 21.0%

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

      12. sqrt-unprod 17.7%

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

      13. add-sqr-sqrt 17.7%

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

   7. Applied egg-rr 17.7%

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

      1. difference-of-squares 17.7%

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

      2. rem-square-sqrt 17.7%

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

      3. rem-square-sqrt 57.0%

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

   9. Simplified 57.0%

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

   10. Taylor expanded in t1 around 0 39.4%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 2.8 \cdot 10^{+195}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. times-frac 97.3%

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

   2. distribute-frac-neg 97.3%

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

   3. distribute-neg-frac2 97.3%

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

   4. +-commutative 97.3%

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

   5. distribute-neg-in 97.3%

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

   6. unsub-neg 97.3%

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

3. Simplified 97.3%

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

   1. frac-2neg 97.3%

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

   2. frac-2neg 97.3%

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

   3. frac-times 68.9%

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

   4. sub-neg 68.9%

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

   5. distribute-neg-in 68.9%

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

   6. +-commutative 68.9%

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

   7. remove-double-neg 68.9%

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

   8. frac-times 97.3%

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

   9. associate-*r/ 98.2%

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

   10. add-sqr-sqrt 50.3%

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

   11. sqrt-unprod 43.5%

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

   12. sqr-neg 43.5%

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

   13. sqrt-unprod 15.5%

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

   14. add-sqr-sqrt 33.0%

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

   15. add-sqr-sqrt 13.9%

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

   16. sqrt-unprod 53.2%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in t1 around inf 57.0%

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

   1. mul-1-neg 57.0%

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

9. Simplified 57.0%

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

10. Final simplification 57.0%

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

Alternative 12: 61.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. times-frac 97.3%

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

   2. distribute-frac-neg 97.3%

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

   3. distribute-neg-frac2 97.3%

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

   4. +-commutative 97.3%

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

   5. distribute-neg-in 97.3%

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

   6. unsub-neg 97.3%

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

3. Simplified 97.3%

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

5. Taylor expanded in t1 around inf 59.9%

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

   1. add-sqr-sqrt 26.2%

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

   2. add-sqr-sqrt 13.8%

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

   3. difference-of-squares 13.8%

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

   4. add-sqr-sqrt 13.8%

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

   5. sqrt-unprod 13.8%

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

   6. sqr-neg 13.8%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 0.0%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 15.1%

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

   11. sqr-neg 15.1%

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

   12. sqrt-unprod 15.0%

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

   13. add-sqr-sqrt 15.0%

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

7. Applied egg-rr 15.0%

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

   1. difference-of-squares 15.0%

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

   2. rem-square-sqrt 29.1%

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

   3. rem-square-sqrt 59.6%

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

9. Simplified 59.6%

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

10. Taylor expanded in v around 0 56.7%

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

Alternative 13: 14.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. times-frac 97.3%

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

   2. distribute-frac-neg 97.3%

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

   3. distribute-neg-frac2 97.3%

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

   4. +-commutative 97.3%

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

   5. distribute-neg-in 97.3%

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

   6. unsub-neg 97.3%

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

3. Simplified 97.3%

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

5. Taylor expanded in t1 around inf 51.8%

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

   1. associate-*r/ 51.8%

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

   2. neg-mul-1 51.8%

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

7. Simplified 51.8%

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

   1. distribute-frac-neg 51.8%

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

   2. div-inv 51.7%

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

   3. distribute-rgt-neg-in 51.7%

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

   4. frac-2neg 51.7%

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

   5. metadata-eval 51.7%

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

   6. add-sqr-sqrt 26.3%

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

   7. sqrt-unprod 25.7%

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

   8. sqr-neg 25.7%

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

   9. sqrt-unprod 5.8%

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

   10. add-sqr-sqrt 11.3%

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

9. Applied egg-rr 11.3%

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

    1. distribute-rgt-neg-out 11.3%

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

    2. *-commutative 11.3%

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

    3. associate-*l/ 11.3%

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

    4. mul-1-neg 11.3%

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

    5. distribute-neg-frac 11.3%

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

    6. remove-double-neg 11.3%

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

11. Simplified 11.3%

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

Reproduce

herbie shell --seed 2024132 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))