Rosa's DopplerBench

Percentage Accurate: 73.1% → 98.1%

Time: 8.2s

Alternatives: 10

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: 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: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(u, v, t1)
	return Float64(Float64(t1 / Float64(t1 + u)) * Float64(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 71.1%

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

   1. associate-/r* 80.5%

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

   2. associate-/l* 96.3%

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

3. Simplified 96.3%

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

4. Taylor expanded in v around 0 80.5%

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

   1. mul-1-neg 80.5%

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

   2. associate-*l/ 97.5%

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

   3. distribute-rgt-neg-out 97.5%

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

6. Simplified 97.5%

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

   1. frac-2neg 97.5%

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

   2. distribute-frac-neg 97.5%

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

   3. add-sqr-sqrt 45.1%

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

   4. sqrt-unprod 49.9%

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

   5. sqr-neg 49.9%

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

   6. sqrt-unprod 20.0%

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

   7. add-sqr-sqrt 36.2%

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

   8. distribute-rgt-neg-in 36.2%

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

   9. frac-2neg 36.2%

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

   10. div-inv 36.2%

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

   11. associate-*l* 36.2%

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

   12. add-sqr-sqrt 20.0%

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

   13. sqrt-unprod 49.8%

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

   14. sqr-neg 49.8%

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

   15. sqrt-unprod 46.5%

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

   16. add-sqr-sqrt 98.7%

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

   17. div-inv 98.9%

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

8. Applied egg-rr 98.9%

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

9. Final simplification 98.9%

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

Alternative 2: 67.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{if}\;u \leq -6 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+39} \lor \neg \left(u \leq 1.2 \cdot 10^{+142}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (/ t1 u) u))))
   (if (<= u -6e+74)
     t_1
     (if (<= u 2.05e+15)
       (- (/ v t1))
       (if (or (<= u 6.5e+39) (not (<= u 1.2e+142)))
         t_1
         (/ (- v) (+ t1 u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * ((t1 / u) / u);
	double tmp;
	if (u <= -6e+74) {
		tmp = t_1;
	} else if (u <= 2.05e+15) {
		tmp = -(v / t1);
	} else if ((u <= 6.5e+39) || !(u <= 1.2e+142)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = v * ((t1 / u) / u)
    if (u <= (-6d+74)) then
        tmp = t_1
    else if (u <= 2.05d+15) then
        tmp = -(v / t1)
    else if ((u <= 6.5d+39) .or. (.not. (u <= 1.2d+142))) then
        tmp = t_1
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * ((t1 / u) / u);
	double tmp;
	if (u <= -6e+74) {
		tmp = t_1;
	} else if (u <= 2.05e+15) {
		tmp = -(v / t1);
	} else if ((u <= 6.5e+39) || !(u <= 1.2e+142)) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * ((t1 / u) / u)
	tmp = 0
	if u <= -6e+74:
		tmp = t_1
	elif u <= 2.05e+15:
		tmp = -(v / t1)
	elif (u <= 6.5e+39) or not (u <= 1.2e+142):
		tmp = t_1
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(t1 / u) / u))
	tmp = 0.0
	if (u <= -6e+74)
		tmp = t_1;
	elseif (u <= 2.05e+15)
		tmp = Float64(-Float64(v / t1));
	elseif ((u <= 6.5e+39) || !(u <= 1.2e+142))
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * ((t1 / u) / u);
	tmp = 0.0;
	if (u <= -6e+74)
		tmp = t_1;
	elseif (u <= 2.05e+15)
		tmp = -(v / t1);
	elseif ((u <= 6.5e+39) || ~((u <= 1.2e+142)))
		tmp = t_1;
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -6e+74], t$95$1, If[LessEqual[u, 2.05e+15], (-N[(v / t1), $MachinePrecision]), If[Or[LessEqual[u, 6.5e+39], N[Not[LessEqual[u, 1.2e+142]], $MachinePrecision]], t$95$1, N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -6e74 or 2.05e15 < u < 6.5000000000000001e39 or 1.2e142 < u

   1. Initial program 77.5%

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

      1. associate-*l/ 75.2%

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

      2. *-commutative 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in t1 around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. neg-mul-1 74.1%

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

      3. unpow2 74.1%

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

   6. Simplified 74.1%

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

      1. associate-/r* 79.5%

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

      2. div-inv 79.4%

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

      3. add-sqr-sqrt 33.8%

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

      4. sqrt-unprod 62.6%

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

      5. sqr-neg 62.6%

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

      6. sqrt-unprod 38.5%

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

      7. add-sqr-sqrt 67.1%

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

   8. Applied egg-rr 67.1%

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

      1. associate-*r/ 67.1%

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

      2. *-rgt-identity 67.1%

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

   10. Simplified 67.1%

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

   if -6e74 < u < 2.05e15

   1. Initial program 65.0%

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

      1. associate-*l/ 73.5%

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

      2. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in t1 around inf 69.6%

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

      1. associate-*r/ 69.6%

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

      2. neg-mul-1 69.6%

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

   6. Simplified 69.6%

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

   if 6.5000000000000001e39 < u < 1.2e142

   1. Initial program 82.0%

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

      1. associate-/r* 89.1%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 54.7%

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

      1. neg-mul-1 54.7%

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

   6. Simplified 54.7%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+74}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+39} \lor \neg \left(u \leq 1.2 \cdot 10^{+142}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 3: 67.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\frac{u}{\frac{t1}{u}}}\\ \mathbf{if}\;u \leq -5.3 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+39}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq 2.6 \cdot 10^{+144}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (/ u (/ t1 u)))))
   (if (<= u -5.3e+73)
     t_1
     (if (<= u 2.4e+15)
       (- (/ v t1))
       (if (<= u 6.5e+39)
         (* v (/ (/ t1 u) u))
         (if (<= u 2.6e+144) (/ (- v) (+ t1 u)) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (u / (t1 / u));
	double tmp;
	if (u <= -5.3e+73) {
		tmp = t_1;
	} else if (u <= 2.4e+15) {
		tmp = -(v / t1);
	} else if (u <= 6.5e+39) {
		tmp = v * ((t1 / u) / u);
	} else if (u <= 2.6e+144) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = v / (u / (t1 / u))
    if (u <= (-5.3d+73)) then
        tmp = t_1
    else if (u <= 2.4d+15) then
        tmp = -(v / t1)
    else if (u <= 6.5d+39) then
        tmp = v * ((t1 / u) / u)
    else if (u <= 2.6d+144) then
        tmp = -v / (t1 + u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (u / (t1 / u));
	double tmp;
	if (u <= -5.3e+73) {
		tmp = t_1;
	} else if (u <= 2.4e+15) {
		tmp = -(v / t1);
	} else if (u <= 6.5e+39) {
		tmp = v * ((t1 / u) / u);
	} else if (u <= 2.6e+144) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (u / (t1 / u))
	tmp = 0
	if u <= -5.3e+73:
		tmp = t_1
	elif u <= 2.4e+15:
		tmp = -(v / t1)
	elif u <= 6.5e+39:
		tmp = v * ((t1 / u) / u)
	elif u <= 2.6e+144:
		tmp = -v / (t1 + u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(u / Float64(t1 / u)))
	tmp = 0.0
	if (u <= -5.3e+73)
		tmp = t_1;
	elseif (u <= 2.4e+15)
		tmp = Float64(-Float64(v / t1));
	elseif (u <= 6.5e+39)
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	elseif (u <= 2.6e+144)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (u / (t1 / u));
	tmp = 0.0;
	if (u <= -5.3e+73)
		tmp = t_1;
	elseif (u <= 2.4e+15)
		tmp = -(v / t1);
	elseif (u <= 6.5e+39)
		tmp = v * ((t1 / u) / u);
	elseif (u <= 2.6e+144)
		tmp = -v / (t1 + u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(u / N[(t1 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -5.3e+73], t$95$1, If[LessEqual[u, 2.4e+15], (-N[(v / t1), $MachinePrecision]), If[LessEqual[u, 6.5e+39], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.6e+144], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -5.29999999999999996e73 or 2.5999999999999999e144 < u

   1. Initial program 75.0%

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

      1. associate-*l/ 73.6%

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

      2. *-commutative 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in t1 around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. neg-mul-1 72.4%

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

      3. unpow2 72.4%

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

   6. Simplified 72.4%

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

      1. clear-num 72.4%

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

      2. un-div-inv 72.4%

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

      3. add-sqr-sqrt 30.6%

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

      4. sqrt-unprod 59.4%

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

      5. sqr-neg 59.4%

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

      6. sqrt-unprod 39.4%

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

      7. add-sqr-sqrt 68.8%

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

   8. Applied egg-rr 68.8%

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

      1. associate-/l* 68.3%

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

      2. add-cbrt-cube 68.9%

         \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt[3]{\left(u \cdot u\right) \cdot u}}}{\frac{t1}{u}}} \]

      3. cbrt-prod 68.8%

         \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt[3]{u \cdot u} \cdot \sqrt[3]{u}}}{\frac{t1}{u}}} \]

      4. *-un-lft-identity 68.8%

         \[\leadsto \frac{v}{\frac{\sqrt[3]{u \cdot u} \cdot \sqrt[3]{u}}{\color{blue}{1 \cdot \frac{t1}{u}}}} \]

      5. times-frac 68.8%

         \[\leadsto \frac{v}{\color{blue}{\frac{\sqrt[3]{u \cdot u}}{1} \cdot \frac{\sqrt[3]{u}}{\frac{t1}{u}}}} \]

      6. cbrt-prod 68.3%

         \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt[3]{u} \cdot \sqrt[3]{u}}}{1} \cdot \frac{\sqrt[3]{u}}{\frac{t1}{u}}} \]

      7. pow2 68.3%

         \[\leadsto \frac{v}{\frac{\color{blue}{{\left(\sqrt[3]{u}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{u}}{\frac{t1}{u}}} \]

   10. Applied egg-rr 68.3%

       \[\leadsto \frac{v}{\color{blue}{\frac{{\left(\sqrt[3]{u}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{u}}{\frac{t1}{u}}}} \]
   11. Step-by-step derivation

       1. /-rgt-identity 68.3%

          \[\leadsto \frac{v}{\color{blue}{{\left(\sqrt[3]{u}\right)}^{2}} \cdot \frac{\sqrt[3]{u}}{\frac{t1}{u}}} \]

       2. associate-*r/ 68.3%

          \[\leadsto \frac{v}{\color{blue}{\frac{{\left(\sqrt[3]{u}\right)}^{2} \cdot \sqrt[3]{u}}{\frac{t1}{u}}}} \]

       3. unpow2 68.3%

          \[\leadsto \frac{v}{\frac{\color{blue}{\left(\sqrt[3]{u} \cdot \sqrt[3]{u}\right)} \cdot \sqrt[3]{u}}{\frac{t1}{u}}} \]

       4. rem-3cbrt-lft 68.3%

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

   12. Simplified 68.3%

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

   if -5.29999999999999996e73 < u < 2.4e15

   1. Initial program 65.0%

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

      1. associate-*l/ 73.5%

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

      2. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in t1 around inf 69.6%

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

      1. associate-*r/ 69.6%

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

      2. neg-mul-1 69.6%

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

   6. Simplified 69.6%

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

   if 2.4e15 < u < 6.5000000000000001e39

   1. Initial program 99.7%

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

      1. associate-*l/ 89.0%

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

      2. *-commutative 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in t1 around 0 89.0%

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

      1. associate-*r/ 89.0%

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

      2. neg-mul-1 89.0%

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

      3. unpow2 89.0%

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

   6. Simplified 89.0%

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

      1. associate-/r* 89.0%

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

      2. div-inv 89.0%

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

      3. add-sqr-sqrt 33.2%

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

      4. sqrt-unprod 67.5%

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

      5. sqr-neg 67.5%

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

      6. sqrt-unprod 34.2%

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

      7. add-sqr-sqrt 56.7%

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

   8. Applied egg-rr 56.7%

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

      1. associate-*r/ 56.7%

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

      2. *-rgt-identity 56.7%

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

   10. Simplified 56.7%

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

   if 6.5000000000000001e39 < u < 2.5999999999999999e144

   1. Initial program 82.0%

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

      1. associate-/r* 89.1%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 54.7%

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

      1. neg-mul-1 54.7%

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

   6. Simplified 54.7%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.3 \cdot 10^{+73}:\\ \;\;\;\;\frac{v}{\frac{u}{\frac{t1}{u}}}\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+39}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq 2.6 \cdot 10^{+144}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\frac{u}{\frac{t1}{u}}}\\ \end{array} \]

Alternative 4: 68.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{if}\;u \leq -5.6 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+39}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (/ (* u u) t1))))
   (if (<= u -5.6e+74)
     t_1
     (if (<= u 1.8e+15)
       (- (/ v t1))
       (if (<= u 6.5e+39)
         (* v (/ (/ t1 u) u))
         (if (<= u 9.5e+144) (/ (- v) (+ t1 u)) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * u) / t1);
	double tmp;
	if (u <= -5.6e+74) {
		tmp = t_1;
	} else if (u <= 1.8e+15) {
		tmp = -(v / t1);
	} else if (u <= 6.5e+39) {
		tmp = v * ((t1 / u) / u);
	} else if (u <= 9.5e+144) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * u) / t1)
    if (u <= (-5.6d+74)) then
        tmp = t_1
    else if (u <= 1.8d+15) then
        tmp = -(v / t1)
    else if (u <= 6.5d+39) then
        tmp = v * ((t1 / u) / u)
    else if (u <= 9.5d+144) then
        tmp = -v / (t1 + u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * u) / t1);
	double tmp;
	if (u <= -5.6e+74) {
		tmp = t_1;
	} else if (u <= 1.8e+15) {
		tmp = -(v / t1);
	} else if (u <= 6.5e+39) {
		tmp = v * ((t1 / u) / u);
	} else if (u <= 9.5e+144) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * u) / t1)
	tmp = 0
	if u <= -5.6e+74:
		tmp = t_1
	elif u <= 1.8e+15:
		tmp = -(v / t1)
	elif u <= 6.5e+39:
		tmp = v * ((t1 / u) / u)
	elif u <= 9.5e+144:
		tmp = -v / (t1 + u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * u) / t1))
	tmp = 0.0
	if (u <= -5.6e+74)
		tmp = t_1;
	elseif (u <= 1.8e+15)
		tmp = Float64(-Float64(v / t1));
	elseif (u <= 6.5e+39)
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	elseif (u <= 9.5e+144)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * u) / t1);
	tmp = 0.0;
	if (u <= -5.6e+74)
		tmp = t_1;
	elseif (u <= 1.8e+15)
		tmp = -(v / t1);
	elseif (u <= 6.5e+39)
		tmp = v * ((t1 / u) / u);
	elseif (u <= 9.5e+144)
		tmp = -v / (t1 + u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -5.6e+74], t$95$1, If[LessEqual[u, 1.8e+15], (-N[(v / t1), $MachinePrecision]), If[LessEqual[u, 6.5e+39], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 9.5e+144], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -5.60000000000000003e74 or 9.50000000000000031e144 < u

   1. Initial program 75.0%

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

      1. associate-*l/ 73.6%

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

      2. *-commutative 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in t1 around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. neg-mul-1 72.4%

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

      3. unpow2 72.4%

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

   6. Simplified 72.4%

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

      1. clear-num 72.4%

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

      2. un-div-inv 72.4%

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

      3. add-sqr-sqrt 30.6%

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

      4. sqrt-unprod 59.4%

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

      5. sqr-neg 59.4%

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

      6. sqrt-unprod 39.4%

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

      7. add-sqr-sqrt 68.8%

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

   8. Applied egg-rr 68.8%

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

   if -5.60000000000000003e74 < u < 1.8e15

   1. Initial program 65.0%

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

      1. associate-*l/ 73.5%

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

      2. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in t1 around inf 69.6%

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

      1. associate-*r/ 69.6%

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

      2. neg-mul-1 69.6%

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

   6. Simplified 69.6%

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

   if 1.8e15 < u < 6.5000000000000001e39

   1. Initial program 99.7%

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

      1. associate-*l/ 89.0%

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

      2. *-commutative 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in t1 around 0 89.0%

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

      1. associate-*r/ 89.0%

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

      2. neg-mul-1 89.0%

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

      3. unpow2 89.0%

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

   6. Simplified 89.0%

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

      1. associate-/r* 89.0%

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

      2. div-inv 89.0%

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

      3. add-sqr-sqrt 33.2%

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

      4. sqrt-unprod 67.5%

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

      5. sqr-neg 67.5%

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

      6. sqrt-unprod 34.2%

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

      7. add-sqr-sqrt 56.7%

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

   8. Applied egg-rr 56.7%

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

      1. associate-*r/ 56.7%

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

      2. *-rgt-identity 56.7%

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

   10. Simplified 56.7%

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

   if 6.5000000000000001e39 < u < 9.50000000000000031e144

   1. Initial program 82.0%

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

      1. associate-/r* 89.1%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 54.7%

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

      1. neg-mul-1 54.7%

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

   6. Simplified 54.7%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+39}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \end{array} \]

Alternative 5: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{+79}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.8e-12)
   (/ (- v) (+ t1 u))
   (if (<= t1 1.45e+79)
     (* (- (/ t1 u)) (/ v u))
     (* (/ v (+ t1 u)) (+ (/ u t1) -1.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.8e-12) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 1.45e+79) {
		tmp = -(t1 / u) * (v / u);
	} else {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.8d-12)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 1.45d+79) then
        tmp = -(t1 / u) * (v / u)
    else
        tmp = (v / (t1 + u)) * ((u / t1) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.8e-12) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 1.45e+79) {
		tmp = -(t1 / u) * (v / u);
	} else {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.8e-12:
		tmp = -v / (t1 + u)
	elif t1 <= 1.45e+79:
		tmp = -(t1 / u) * (v / u)
	else:
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.8e-12)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 1.45e+79)
		tmp = Float64(Float64(-Float64(t1 / u)) * Float64(v / u));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(u / t1) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.8e-12)
		tmp = -v / (t1 + u);
	elseif (t1 <= 1.45e+79)
		tmp = -(t1 / u) * (v / u);
	else
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.8e-12], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.45e+79], N[((-N[(t1 / u), $MachinePrecision]) * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.8000000000000002e-12

   1. Initial program 60.8%

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

      1. associate-/r* 72.4%

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

      2. associate-/l* 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around inf 83.7%

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

      1. neg-mul-1 83.7%

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

   6. Simplified 83.7%

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

   if -2.8000000000000002e-12 < t1 < 1.44999999999999996e79

   1. Initial program 84.3%

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

      1. associate-/r* 89.1%

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

      2. associate-/l* 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in v around 0 89.1%

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

      1. mul-1-neg 89.1%

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

      2. associate-*l/ 95.6%

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

      3. distribute-rgt-neg-out 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in t1 around 0 70.0%

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

      1. associate-*r/ 70.0%

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

      2. associate-*r* 70.0%

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

      3. unpow2 70.0%

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

      4. times-frac 78.4%

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

      5. mul-1-neg 78.4%

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

   9. Simplified 78.4%

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

   if 1.44999999999999996e79 < t1

   1. Initial program 45.8%

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

      1. associate-/r* 65.6%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 79.6%

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

      1. neg-mul-1 79.6%

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

      2. +-commutative 79.6%

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

      3. unsub-neg 79.6%

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

      4. *-commutative 79.6%

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

      5. associate-/l* 85.3%

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

   6. Simplified 85.3%

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

   7. Taylor expanded in v around 0 85.3%

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

      1. *-rgt-identity 85.3%

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

      2. associate-*r/ 85.1%

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

      3. *-commutative 85.1%

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

      4. associate-*l* 86.8%

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

      5. sub-neg 86.8%

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

      6. metadata-eval 86.8%

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

      7. associate-*r/ 87.0%

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

      8. *-rgt-identity 87.0%

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

   9. Simplified 87.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{+79}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \end{array} \]

Alternative 6: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-96} \lor \neg \left(t1 \leq 10^{-14}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.9e-96) (not (<= t1 1e-14)))
   (/ (- v) (+ t1 u))
   (/ (- v) (* u (/ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.9e-96) || !(t1 <= 1e-14)) {
		tmp = -v / (t1 + u);
	} 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 <= (-2.9d-96)) .or. (.not. (t1 <= 1d-14))) then
        tmp = -v / (t1 + u)
    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 <= -2.9e-96) || !(t1 <= 1e-14)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -v / (u * (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.9e-96) or not (t1 <= 1e-14):
		tmp = -v / (t1 + u)
	else:
		tmp = -v / (u * (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.9e-96) || !(t1 <= 1e-14))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(-v) / Float64(u * Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.9e-96) || ~((t1 <= 1e-14)))
		tmp = -v / (t1 + u);
	else
		tmp = -v / (u * (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.9e-96], N[Not[LessEqual[t1, 1e-14]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.89999999999999994e-96 or 9.99999999999999999e-15 < t1

   1. Initial program 62.2%

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

      1. associate-/r* 74.6%

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

      2. associate-/l* 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in t1 around inf 78.9%

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

      1. neg-mul-1 78.9%

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

   6. Simplified 78.9%

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

   if -2.89999999999999994e-96 < t1 < 9.99999999999999999e-15

   1. Initial program 81.5%

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

      1. associate-/r* 87.4%

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

      2. associate-/l* 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in v around 0 87.4%

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

      1. mul-1-neg 87.4%

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

      2. associate-*l/ 94.6%

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

      3. distribute-rgt-neg-out 94.6%

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

   6. Simplified 94.6%

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

      1. frac-2neg 94.6%

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

      2. distribute-frac-neg 94.6%

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

      3. add-sqr-sqrt 40.6%

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

      4. sqrt-unprod 46.7%

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

      5. sqr-neg 46.7%

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

      6. sqrt-unprod 21.7%

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

      7. add-sqr-sqrt 40.2%

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

      8. distribute-rgt-neg-in 40.2%

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

      9. frac-2neg 40.2%

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

      10. div-inv 40.2%

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

      11. associate-*l* 40.1%

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

      12. add-sqr-sqrt 21.7%

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

      13. sqrt-unprod 46.7%

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

      14. sqr-neg 46.7%

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

      15. sqrt-unprod 43.7%

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

      16. add-sqr-sqrt 97.6%

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

      17. div-inv 97.7%

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

   8. Applied egg-rr 97.7%

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

      1. div-inv 97.6%

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

      2. add-cube-cbrt 96.9%

         \[\leadsto -\frac{t1}{t1 + u} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right) \cdot \sqrt[3]{v}\right)} \cdot \frac{1}{t1 + u}\right) \]

      3. associate-*l* 96.9%

         \[\leadsto -\frac{t1}{t1 + u} \cdot \color{blue}{\left(\left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right) \cdot \left(\sqrt[3]{v} \cdot \frac{1}{t1 + u}\right)\right)} \]

      4. pow2 96.9%

         \[\leadsto -\frac{t1}{t1 + u} \cdot \left(\color{blue}{{\left(\sqrt[3]{v}\right)}^{2}} \cdot \left(\sqrt[3]{v} \cdot \frac{1}{t1 + u}\right)\right) \]

   10. Applied egg-rr 96.9%

       \[\leadsto -\frac{t1}{t1 + u} \cdot \color{blue}{\left({\left(\sqrt[3]{v}\right)}^{2} \cdot \left(\sqrt[3]{v} \cdot \frac{1}{t1 + u}\right)\right)} \]

   11. Taylor expanded in t1 around 0 73.7%

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

       1. *-commutative 73.7%

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

       2. associate-*r/ 74.4%

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

       3. unpow2 74.4%

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

       4. *-lft-identity 74.4%

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

       5. unpow2 74.4%

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

       6. associate-*l/ 73.9%

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

       7. associate-/r/ 74.3%

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

       8. unpow2 74.3%

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

       9. associate-*l/ 82.0%

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

       10. associate-*r/ 82.0%

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

       11. *-commutative 82.0%

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

       12. *-rgt-identity 82.0%

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

   13. Simplified 82.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-96} \lor \neg \left(t1 \leq 10^{-14}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \end{array} \]

Alternative 7: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.9 \cdot 10^{-13} \lor \neg \left(t1 \leq 1.25 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.9e-13) (not (<= t1 1.25e-14)))
   (/ (- v) (+ t1 u))
   (* (- (/ t1 u)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.9e-13) || !(t1 <= 1.25e-14)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -(t1 / u) * (v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3.9d-13)) .or. (.not. (t1 <= 1.25d-14))) then
        tmp = -v / (t1 + u)
    else
        tmp = -(t1 / u) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.9e-13) || !(t1 <= 1.25e-14)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -(t1 / u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.9e-13) or not (t1 <= 1.25e-14):
		tmp = -v / (t1 + u)
	else:
		tmp = -(t1 / u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.9e-13) || !(t1 <= 1.25e-14))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(-Float64(t1 / u)) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.9e-13) || ~((t1 <= 1.25e-14)))
		tmp = -v / (t1 + u);
	else
		tmp = -(t1 / u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.9e-13], N[Not[LessEqual[t1, 1.25e-14]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-N[(t1 / u), $MachinePrecision]) * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.90000000000000004e-13 or 1.25e-14 < t1

   1. Initial program 58.3%

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

      1. associate-/r* 72.0%

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

      2. associate-/l* 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in t1 around inf 81.4%

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

      1. neg-mul-1 81.4%

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

   6. Simplified 81.4%

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

   if -3.90000000000000004e-13 < t1 < 1.25e-14

   1. Initial program 83.3%

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

      1. associate-/r* 88.6%

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

      2. associate-/l* 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in v around 0 88.6%

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

      1. mul-1-neg 88.6%

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

      2. associate-*l/ 95.2%

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

      3. distribute-rgt-neg-out 95.2%

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

   6. Simplified 95.2%

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

   7. Taylor expanded in t1 around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. associate-*r* 71.9%

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

      3. unpow2 71.9%

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

      4. times-frac 81.2%

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

      5. mul-1-neg 81.2%

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

   9. Simplified 81.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.9 \cdot 10^{-13} \lor \neg \left(t1 \leq 1.25 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \end{array} \]

Alternative 8: 63.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. associate-/r* 80.5%

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

   2. associate-/l* 96.3%

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

3. Simplified 96.3%

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

4. Taylor expanded in t1 around inf 56.3%

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

   1. neg-mul-1 56.3%

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

6. Simplified 56.3%

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

7. Final simplification 56.3%

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

Alternative 9: 55.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

   1. associate-*l/ 74.2%

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

   2. *-commutative 74.2%

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

3. Simplified 74.2%

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

4. Taylor expanded in t1 around inf 50.5%

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

   1. associate-*r/ 50.5%

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

   2. neg-mul-1 50.5%

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

6. Simplified 50.5%

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

7. Final simplification 50.5%

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

Alternative 10: 15.1% 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 71.1%

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

   1. associate-*l/ 74.2%

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

   2. *-commutative 74.2%

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

3. Simplified 74.2%

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

   1. associate-/r* 95.7%

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

   2. associate-*r/ 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-/r/ 96.3%

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

   5. div-inv 96.2%

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

   6. frac-2neg 96.2%

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

   7. frac-times 82.4%

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

   8. remove-double-neg 82.4%

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

   9. *-commutative 82.4%

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

   10. *-un-lft-identity 82.4%

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

   11. distribute-neg-frac 82.4%

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

   12. distribute-neg-in 82.4%

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

   13. add-sqr-sqrt 39.8%

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

   14. sqrt-unprod 67.7%

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

   15. sqr-neg 67.7%

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

   16. sqrt-unprod 31.8%

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

   17. add-sqr-sqrt 59.2%

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

   18. sub-neg 59.2%

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

5. Applied egg-rr 59.2%

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

6. Taylor expanded in t1 around inf 14.0%

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

7. Final simplification 14.0%

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

Reproduce

herbie shell --seed 2023264 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))