Rosa's DopplerBench

Percentage Accurate: 72.6% → 98.0%

Time: 12.6s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.6% 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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. associate-/l* 73.0%

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

   2. distribute-lft-neg-out 73.0%

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

   3. distribute-rgt-neg-in 73.0%

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

   4. associate-/r* 82.0%

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

   5. distribute-neg-frac2 82.0%

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

3. Simplified 82.0%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

   3. distribute-neg-in 97.4%

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

   4. sub-neg 97.4%

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

   5. associate-*l/ 97.8%

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

   6. frac-2neg 97.8%

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

   7. associate-*r/ 98.3%

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

   8. remove-double-neg 98.3%

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

   9. sub-neg 98.3%

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

   10. distribute-neg-in 98.3%

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

   11. +-commutative 98.3%

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

   12. frac-2neg 98.3%

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

   13. add-sqr-sqrt 50.6%

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

   14. sqrt-unprod 42.4%

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

   15. sqr-neg 42.4%

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

   16. sqrt-unprod 19.5%

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

   17. add-sqr-sqrt 40.8%

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

   18. add-sqr-sqrt 17.8%

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

   19. sqrt-unprod 58.5%

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

6. Applied egg-rr 98.3%

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

Alternative 2: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{v}{t1}\\ \mathbf{if}\;t1 \leq -7.5 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.8 \cdot 10^{+157}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* -1.0 (/ v t1))))
   (if (<= t1 -7.5e+168)
     t_1
     (if (<= t1 1.8e+157) (* t1 (/ (/ v (+ t1 u)) (- (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -1.0 * (v / t1);
	double tmp;
	if (t1 <= -7.5e+168) {
		tmp = t_1;
	} else if (t1 <= 1.8e+157) {
		tmp = t1 * ((v / (t1 + u)) / -(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 = (-1.0d0) * (v / t1)
    if (t1 <= (-7.5d+168)) then
        tmp = t_1
    else if (t1 <= 1.8d+157) then
        tmp = t1 * ((v / (t1 + u)) / -(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 = -1.0 * (v / t1);
	double tmp;
	if (t1 <= -7.5e+168) {
		tmp = t_1;
	} else if (t1 <= 1.8e+157) {
		tmp = t1 * ((v / (t1 + u)) / -(t1 + u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -1.0 * (v / t1)
	tmp = 0
	if t1 <= -7.5e+168:
		tmp = t_1
	elif t1 <= 1.8e+157:
		tmp = t1 * ((v / (t1 + u)) / -(t1 + u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-1.0 * Float64(v / t1))
	tmp = 0.0
	if (t1 <= -7.5e+168)
		tmp = t_1;
	elseif (t1 <= 1.8e+157)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(-Float64(t1 + u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -1.0 * (v / t1);
	tmp = 0.0;
	if (t1 <= -7.5e+168)
		tmp = t_1;
	elseif (t1 <= 1.8e+157)
		tmp = t1 * ((v / (t1 + u)) / -(t1 + u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(-1.0 * N[(v / t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.5e+168], t$95$1, If[LessEqual[t1, 1.8e+157], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / (-N[(t1 + u), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.4999999999999999e168 or 1.80000000000000012e157 < t1

   1. Initial program 42.5%

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

      1. associate-/l* 43.6%

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

      2. distribute-lft-neg-out 43.6%

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

      3. distribute-rgt-neg-in 43.6%

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

      4. associate-/r* 57.0%

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

      5. distribute-neg-frac2 57.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in t1 around inf 87.4%

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

   if -7.4999999999999999e168 < t1 < 1.80000000000000012e157

   1. Initial program 83.0%

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

      1. associate-/l* 84.5%

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

      2. distribute-lft-neg-out 84.5%

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

      3. distribute-rgt-neg-in 84.5%

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

      4. associate-/r* 91.8%

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

      5. distribute-neg-frac2 91.8%

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

   3. Simplified 91.8%

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

Alternative 3: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1 + u} \cdot \frac{t1}{u}\\ \mathbf{if}\;u \leq -5.8 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.2 \cdot 10^{-74}:\\ \;\;\;\;-1 \cdot \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (* (/ v (+ t1 u)) (/ t1 u)))))
   (if (<= u -5.8e+56) t_1 (if (<= u 1.2e-74) (* -1.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -((v / (t1 + u)) * (t1 / u));
	double tmp;
	if (u <= -5.8e+56) {
		tmp = t_1;
	} else if (u <= 1.2e-74) {
		tmp = -1.0 * (v / t1);
	} 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 / (t1 + u)) * (t1 / u))
    if (u <= (-5.8d+56)) then
        tmp = t_1
    else if (u <= 1.2d-74) then
        tmp = (-1.0d0) * (v / t1)
    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 / (t1 + u)) * (t1 / u));
	double tmp;
	if (u <= -5.8e+56) {
		tmp = t_1;
	} else if (u <= 1.2e-74) {
		tmp = -1.0 * (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -((v / (t1 + u)) * (t1 / u))
	tmp = 0
	if u <= -5.8e+56:
		tmp = t_1
	elif u <= 1.2e-74:
		tmp = -1.0 * (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / u)))
	tmp = 0.0
	if (u <= -5.8e+56)
		tmp = t_1;
	elseif (u <= 1.2e-74)
		tmp = Float64(-1.0 * Float64(v / t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -((v / (t1 + u)) * (t1 / u));
	tmp = 0.0;
	if (u <= -5.8e+56)
		tmp = t_1;
	elseif (u <= 1.2e-74)
		tmp = -1.0 * (v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[u, -5.8e+56], t$95$1, If[LessEqual[u, 1.2e-74], N[(-1.0 * N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -5.80000000000000014e56 or 1.1999999999999999e-74 < u

   1. Initial program 79.8%

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

      1. associate-/l* 81.1%

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

      2. distribute-lft-neg-out 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

      4. associate-/r* 90.6%

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

      5. distribute-neg-frac2 90.6%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in t1 around 0 86.8%

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

      1. distribute-frac-neg2 86.8%

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

      2. add-sqr-sqrt 52.6%

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

      3. sqrt-unprod 75.3%

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

      4. sqr-neg 75.3%

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

      5. sqrt-unprod 26.9%

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

      6. add-sqr-sqrt 63.8%

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

      7. associate-/l/ 64.0%

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

      8. add-sqr-sqrt 26.9%

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

      9. sqrt-unprod 75.3%

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

      10. sqr-neg 75.3%

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

      11. sqrt-unprod 48.3%

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

      12. add-sqr-sqrt 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. distribute-rgt-neg-out 78.9%

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

      2. associate-*r/ 77.6%

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

      3. associate-/l/ 82.2%

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

      4. div-inv 82.1%

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

      5. add-sqr-sqrt 54.3%

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

      6. sqrt-unprod 66.3%

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

      7. sqr-neg 66.3%

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

      8. mul-1-neg 66.3%

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

      9. mul-1-neg 66.3%

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

      10. sqrt-unprod 50.8%

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

      11. add-sqr-sqrt 58.0%

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

      12. mul-1-neg 58.0%

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

      13. distribute-neg-frac 58.0%

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

      14. associate-*l/ 58.0%

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

   9. Applied egg-rr 82.2%

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

       1. associate-*r/ 92.9%

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

       2. *-commutative 92.9%

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

       3. associate-/l* 91.9%

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

   11. Applied egg-rr 91.9%

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

   if -5.80000000000000014e56 < u < 1.1999999999999999e-74

   1. Initial program 63.2%

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

      1. associate-/l* 64.6%

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

      2. distribute-lft-neg-out 64.6%

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

      3. distribute-rgt-neg-in 64.6%

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

      4. associate-/r* 73.1%

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

      5. distribute-neg-frac2 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in t1 around inf 79.4%

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

Alternative 4: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;-\frac{\frac{t1 \cdot v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -7.5e-37)
     t_1
     (if (<= t1 1.9e-10) (- (/ (/ (* t1 v) u) u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -7.5e-37) {
		tmp = t_1;
	} else if (t1 <= 1.9e-10) {
		tmp = -(((t1 * v) / u) / 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 / (t1 + u)
    if (t1 <= (-7.5d-37)) then
        tmp = t_1
    else if (t1 <= 1.9d-10) then
        tmp = -(((t1 * v) / u) / 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 / (t1 + u);
	double tmp;
	if (t1 <= -7.5e-37) {
		tmp = t_1;
	} else if (t1 <= 1.9e-10) {
		tmp = -(((t1 * v) / u) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -7.5e-37:
		tmp = t_1
	elif t1 <= 1.9e-10:
		tmp = -(((t1 * v) / u) / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -7.5e-37)
		tmp = t_1;
	elseif (t1 <= 1.9e-10)
		tmp = Float64(-Float64(Float64(Float64(t1 * v) / u) / u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -7.5e-37)
		tmp = t_1;
	elseif (t1 <= 1.9e-10)
		tmp = -(((t1 * v) / u) / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.5e-37], t$95$1, If[LessEqual[t1, 1.9e-10], (-N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] / u), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.5000000000000004e-37 or 1.8999999999999999e-10 < t1

   1. Initial program 56.6%

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

      1. associate-/l* 59.0%

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

      2. distribute-lft-neg-out 59.0%

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

      3. distribute-rgt-neg-in 59.0%

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

      4. associate-/r* 73.2%

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

      5. distribute-neg-frac2 73.2%

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

   3. Simplified 73.2%

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

      1. associate-*r/ 99.9%

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

      2. frac-2neg 99.9%

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

      3. remove-double-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 82.9%

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

   if -7.5000000000000004e-37 < t1 < 1.8999999999999999e-10

   1. Initial program 86.9%

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

      1. associate-/l* 87.2%

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

      2. distribute-lft-neg-out 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. associate-/r* 90.9%

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

      5. distribute-neg-frac2 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in t1 around 0 79.3%

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

      1. distribute-frac-neg2 79.3%

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

      2. add-sqr-sqrt 42.0%

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

      3. sqrt-unprod 60.5%

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

      4. sqr-neg 60.5%

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

      5. sqrt-unprod 19.2%

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

      6. add-sqr-sqrt 48.0%

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

      7. associate-/l/ 48.0%

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

      8. add-sqr-sqrt 19.2%

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

      9. sqrt-unprod 60.5%

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

      10. sqr-neg 60.5%

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

      11. sqrt-unprod 41.2%

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

      12. add-sqr-sqrt 77.1%

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

   7. Applied egg-rr 77.1%

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

      1. distribute-rgt-neg-out 77.1%

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

      2. associate-*r/ 76.4%

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

      3. associate-/l/ 79.6%

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

      4. div-inv 79.5%

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

      5. add-sqr-sqrt 58.0%

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

      6. sqrt-unprod 64.3%

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

      7. sqr-neg 64.3%

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

      8. mul-1-neg 64.3%

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

      9. mul-1-neg 64.3%

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

      10. sqrt-unprod 43.5%

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

      11. add-sqr-sqrt 48.2%

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

      12. mul-1-neg 48.2%

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

      13. distribute-neg-frac 48.2%

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

      14. associate-*l/ 48.2%

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

   9. Applied egg-rr 79.6%

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

   10. Taylor expanded in t1 around 0 82.2%

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

Alternative 5: 57.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{u}\\ \mathbf{if}\;u \leq -3.1 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+157}:\\ \;\;\;\;-1 \cdot \frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v u))))
   (if (<= u -3.1e+136) t_1 (if (<= u 9.5e+157) (* -1.0 (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / u);
	double tmp;
	if (u <= -3.1e+136) {
		tmp = t_1;
	} else if (u <= 9.5e+157) {
		tmp = -1.0 * (v / t1);
	} 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)
    if (u <= (-3.1d+136)) then
        tmp = t_1
    else if (u <= 9.5d+157) then
        tmp = (-1.0d0) * (v / t1)
    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);
	double tmp;
	if (u <= -3.1e+136) {
		tmp = t_1;
	} else if (u <= 9.5e+157) {
		tmp = -1.0 * (v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / u)
	tmp = 0
	if u <= -3.1e+136:
		tmp = t_1
	elif u <= 9.5e+157:
		tmp = -1.0 * (v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / u))
	tmp = 0.0
	if (u <= -3.1e+136)
		tmp = t_1;
	elseif (u <= 9.5e+157)
		tmp = Float64(-1.0 * Float64(v / t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(v / u);
	tmp = 0.0;
	if (u <= -3.1e+136)
		tmp = t_1;
	elseif (u <= 9.5e+157)
		tmp = -1.0 * (v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / u), $MachinePrecision])}, If[LessEqual[u, -3.1e+136], t$95$1, If[LessEqual[u, 9.5e+157], N[(-1.0 * N[(v / t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.09999999999999983e136 or 9.4999999999999996e157 < u

   1. Initial program 74.9%

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

      1. associate-/l* 75.3%

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

      2. distribute-lft-neg-out 75.3%

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

      3. distribute-rgt-neg-in 75.3%

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

      4. associate-/r* 88.3%

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

      5. distribute-neg-frac2 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in t1 around 0 88.0%

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

   6. Taylor expanded in u around 0 48.2%

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

      1. mul-1-neg 48.2%

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

      2. distribute-rgt-neg-out 48.2%

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

      3. associate-/r* 47.8%

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

   8. Applied egg-rr 47.8%

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

   9. Taylor expanded in t1 around 0 45.5%

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

   if -3.09999999999999983e136 < u < 9.4999999999999996e157

   1. Initial program 70.1%

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

      1. associate-/l* 71.9%

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

      2. distribute-lft-neg-out 71.9%

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

      3. distribute-rgt-neg-in 71.9%

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

      4. associate-/r* 79.1%

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

      5. distribute-neg-frac2 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in t1 around inf 65.4%

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

Alternative 6: 23.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.7 \cdot 10^{+121}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 2.05 \cdot 10^{+62}:\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -7.7e+121) (/ v t1) (if (<= t1 2.05e+62) (- (/ v u)) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -7.7e+121) {
		tmp = v / t1;
	} else if (t1 <= 2.05e+62) {
		tmp = -(v / u);
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-7.7d+121)) then
        tmp = v / t1
    else if (t1 <= 2.05d+62) then
        tmp = -(v / u)
    else
        tmp = v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -7.7e+121) {
		tmp = v / t1;
	} else if (t1 <= 2.05e+62) {
		tmp = -(v / u);
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -7.7e+121:
		tmp = v / t1
	elif t1 <= 2.05e+62:
		tmp = -(v / u)
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -7.7e+121)
		tmp = Float64(v / t1);
	elseif (t1 <= 2.05e+62)
		tmp = Float64(-Float64(v / u));
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -7.7e+121)
		tmp = v / t1;
	elseif (t1 <= 2.05e+62)
		tmp = -(v / u);
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -7.7e+121], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 2.05e+62], (-N[(v / u), $MachinePrecision]), N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.70000000000000028e121 or 2.04999999999999992e62 < t1

   1. Initial program 46.5%

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

      1. times-frac 99.9%

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

      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 85.7%

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

   6. Taylor expanded in u around inf 39.0%

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

   if -7.70000000000000028e121 < t1 < 2.04999999999999992e62

   1. Initial program 85.7%

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

      1. associate-/l* 86.1%

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

      2. distribute-lft-neg-out 86.1%

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

      3. distribute-rgt-neg-in 86.1%

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

      4. associate-/r* 91.3%

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

      5. distribute-neg-frac2 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in t1 around 0 71.2%

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

   6. Taylor expanded in u around 0 26.3%

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

      1. mul-1-neg 26.3%

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

      2. distribute-rgt-neg-out 26.3%

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

      3. associate-/r* 26.8%

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

   8. Applied egg-rr 26.8%

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

   9. Taylor expanded in t1 around 0 26.3%

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

Alternative 7: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. times-frac 97.8%

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

   2. distribute-frac-neg 97.8%

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

   3. distribute-neg-frac2 97.8%

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

   4. +-commutative 97.8%

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

   5. distribute-neg-in 97.8%

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

   6. unsub-neg 97.8%

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

3. Simplified 97.8%

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

Alternative 8: 61.2% 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 71.6%

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

   1. associate-/l* 73.0%

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

   2. distribute-lft-neg-out 73.0%

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

   3. distribute-rgt-neg-in 73.0%

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

   4. associate-/r* 82.0%

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

   5. distribute-neg-frac2 82.0%

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

3. Simplified 82.0%

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

   1. associate-*r/ 97.4%

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

   2. frac-2neg 97.4%

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

   3. remove-double-neg 97.4%

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

6. Applied egg-rr 97.4%

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

7. Taylor expanded in t1 around inf 61.9%

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

Alternative 9: 14.1% 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 71.6%

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

   1. times-frac 97.8%

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

   2. distribute-frac-neg 97.8%

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

   3. distribute-neg-frac2 97.8%

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

   4. +-commutative 97.8%

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

   5. distribute-neg-in 97.8%

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

   6. unsub-neg 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in t1 around inf 52.4%

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

6. Taylor expanded in u around inf 16.2%

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

Reproduce

herbie shell --seed 2024076 -o generate:simplify
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))