Rosa's DopplerBench

Percentage Accurate: 73.2% → 97.8%

Time: 7.9s

Alternatives: 11

Speedup: 0.9×

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: 73.2% 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: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-frac-neg2 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-neg.f64 99.0

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1}\\ t_2 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -6 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -3.6 \cdot 10^{-112}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 9.2 \cdot 10^{-222}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) t1)) (t_2 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -6e+116)
     t_1
     (if (<= t1 -3.6e-112)
       t_2
       (if (<= t1 9.2e-222)
         (* (/ (- v) u) (/ t1 u))
         (if (<= t1 2.05e+102) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / t1;
	double t_2 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -6e+116) {
		tmp = t_1;
	} else if (t1 <= -3.6e-112) {
		tmp = t_2;
	} else if (t1 <= 9.2e-222) {
		tmp = (-v / u) * (t1 / u);
	} else if (t1 <= 2.05e+102) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = -v / t1
    t_2 = (-t1 * v) / ((t1 + u) * (t1 + u))
    if (t1 <= (-6d+116)) then
        tmp = t_1
    else if (t1 <= (-3.6d-112)) then
        tmp = t_2
    else if (t1 <= 9.2d-222) then
        tmp = (-v / u) * (t1 / u)
    else if (t1 <= 2.05d+102) then
        tmp = t_2
    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;
	double t_2 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -6e+116) {
		tmp = t_1;
	} else if (t1 <= -3.6e-112) {
		tmp = t_2;
	} else if (t1 <= 9.2e-222) {
		tmp = (-v / u) * (t1 / u);
	} else if (t1 <= 2.05e+102) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / t1
	t_2 = (-t1 * v) / ((t1 + u) * (t1 + u))
	tmp = 0
	if t1 <= -6e+116:
		tmp = t_1
	elif t1 <= -3.6e-112:
		tmp = t_2
	elif t1 <= 9.2e-222:
		tmp = (-v / u) * (t1 / u)
	elif t1 <= 2.05e+102:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / t1)
	t_2 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -6e+116)
		tmp = t_1;
	elseif (t1 <= -3.6e-112)
		tmp = t_2;
	elseif (t1 <= 9.2e-222)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	elseif (t1 <= 2.05e+102)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / t1;
	t_2 = (-t1 * v) / ((t1 + u) * (t1 + u));
	tmp = 0.0;
	if (t1 <= -6e+116)
		tmp = t_1;
	elseif (t1 <= -3.6e-112)
		tmp = t_2;
	elseif (t1 <= 9.2e-222)
		tmp = (-v / u) * (t1 / u);
	elseif (t1 <= 2.05e+102)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / t1), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -6e+116], t$95$1, If[LessEqual[t1, -3.6e-112], t$95$2, If[LessEqual[t1, 9.2e-222], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.05e+102], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.9999999999999997e116 or 2.05e102 < t1

   1. Initial program 45.0%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if -5.9999999999999997e116 < t1 < -3.6000000000000001e-112 or 9.2000000000000005e-222 < t1 < 2.05e102

   1. Initial program 95.4%

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

   if -3.6000000000000001e-112 < t1 < 9.2000000000000005e-222

   1. Initial program 78.8%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{+116}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -3.6 \cdot 10^{-112}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 9.2 \cdot 10^{-222}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) -2e-16)
   (/ (- v) t1)
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (((-t1 * v) / ((t1 + u) * (t1 + u))) <= -2e-16) {
		tmp = -v / t1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (((-t1 * v) / ((t1 + u) * (t1 + u))) <= (-2d-16)) then
        tmp = -v / t1
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (((-t1 * v) / ((t1 + u) * (t1 + u))) <= -2e-16) {
		tmp = -v / t1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if ((-t1 * v) / ((t1 + u) * (t1 + u))) <= -2e-16:
		tmp = -v / t1
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u))) <= -2e-16)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (((-t1 * v) / ((t1 + u) * (t1 + u))) <= -2e-16)
		tmp = -v / t1;
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-16], N[((-v) / t1), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u))) < -2e-16

   1. Initial program 90.5%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -2e-16 < (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u)))

   1. Initial program 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-neg.f64 98.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-neg.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 98.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 98.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 98.8

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

   6. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. neg-mul-1 N/A

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

      8. lift-neg.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 97.6%

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

   9. Taylor expanded in u around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 59.3

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

   11. Applied rewrites 59.3%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -8.8 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-51}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -8.8e-72)
     t_1
     (if (<= t1 1.26e-51) (* (/ (- v) u) (/ t1 u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -8.8e-72) {
		tmp = t_1;
	} else if (t1 <= 1.26e-51) {
		tmp = (-v / 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 = -v / (t1 + u)
    if (t1 <= (-8.8d-72)) then
        tmp = t_1
    else if (t1 <= 1.26d-51) then
        tmp = (-v / 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 = -v / (t1 + u);
	double tmp;
	if (t1 <= -8.8e-72) {
		tmp = t_1;
	} else if (t1 <= 1.26e-51) {
		tmp = (-v / u) * (t1 / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -8.8e-72:
		tmp = t_1
	elif t1 <= 1.26e-51:
		tmp = (-v / u) * (t1 / 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 <= -8.8e-72)
		tmp = t_1;
	elseif (t1 <= 1.26e-51)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / 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 <= -8.8e-72)
		tmp = t_1;
	elseif (t1 <= 1.26e-51)
		tmp = (-v / u) * (t1 / 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, -8.8e-72], t$95$1, If[LessEqual[t1, 1.26e-51], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.8000000000000001e-72 or 1.2600000000000001e-51 < t1

   1. Initial program 68.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-neg.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-neg.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 99.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. neg-mul-1 N/A

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

      8. lift-neg.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in u around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 83.7

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

   11. Applied rewrites 83.7%

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

   if -8.8000000000000001e-72 < t1 < 1.2600000000000001e-51

   1. Initial program 86.5%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 91.2

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

   5. Applied rewrites 91.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-51}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -8.8 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-51}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -8.8e-72)
     t_1
     (if (<= t1 1.26e-51) (* (/ (- v) (* u u)) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -8.8e-72) {
		tmp = t_1;
	} else if (t1 <= 1.26e-51) {
		tmp = (-v / (u * u)) * 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)
    if (t1 <= (-8.8d-72)) then
        tmp = t_1
    else if (t1 <= 1.26d-51) then
        tmp = (-v / (u * u)) * 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);
	double tmp;
	if (t1 <= -8.8e-72) {
		tmp = t_1;
	} else if (t1 <= 1.26e-51) {
		tmp = (-v / (u * u)) * t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -8.8e-72:
		tmp = t_1
	elif t1 <= 1.26e-51:
		tmp = (-v / (u * u)) * t1
	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 <= -8.8e-72)
		tmp = t_1;
	elseif (t1 <= 1.26e-51)
		tmp = Float64(Float64(Float64(-v) / Float64(u * u)) * t1);
	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 <= -8.8e-72)
		tmp = t_1;
	elseif (t1 <= 1.26e-51)
		tmp = (-v / (u * u)) * t1;
	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, -8.8e-72], t$95$1, If[LessEqual[t1, 1.26e-51], N[(N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.8000000000000001e-72 or 1.2600000000000001e-51 < t1

   1. Initial program 68.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-neg.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-neg.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 99.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. neg-mul-1 N/A

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

      8. lift-neg.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in u around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 83.7

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

   11. Applied rewrites 83.7%

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

   if -8.8000000000000001e-72 < t1 < 1.2600000000000001e-51

   1. Initial program 86.5%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.8

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

   7. Applied rewrites 84.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-51}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -8.8 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-51}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -8.8e-72)
     t_1
     (if (<= t1 1.26e-51) (* (/ (- t1) (* u u)) v) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -8.8e-72) {
		tmp = t_1;
	} else if (t1 <= 1.26e-51) {
		tmp = (-t1 / (u * u)) * v;
	} 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 <= (-8.8d-72)) then
        tmp = t_1
    else if (t1 <= 1.26d-51) then
        tmp = (-t1 / (u * u)) * v
    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 <= -8.8e-72) {
		tmp = t_1;
	} else if (t1 <= 1.26e-51) {
		tmp = (-t1 / (u * u)) * v;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -8.8e-72:
		tmp = t_1
	elif t1 <= 1.26e-51:
		tmp = (-t1 / (u * u)) * v
	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 <= -8.8e-72)
		tmp = t_1;
	elseif (t1 <= 1.26e-51)
		tmp = Float64(Float64(Float64(-t1) / Float64(u * u)) * v);
	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 <= -8.8e-72)
		tmp = t_1;
	elseif (t1 <= 1.26e-51)
		tmp = (-t1 / (u * u)) * v;
	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, -8.8e-72], t$95$1, If[LessEqual[t1, 1.26e-51], N[(N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.8000000000000001e-72 or 1.2600000000000001e-51 < t1

   1. Initial program 68.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-neg.f64 99.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-neg.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 99.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. neg-mul-1 N/A

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

      8. lift-neg.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in u around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 83.7

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

   11. Applied rewrites 83.7%

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

   if -8.8000000000000001e-72 < t1 < 1.2600000000000001e-51

   1. Initial program 86.5%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.0

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

   7. Applied rewrites 83.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-51}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 0.8× speedup

Math

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

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)) * v) / Float64(-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 76.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-neg-in N/A

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 97.9

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 97.9

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 97.9

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

4. Applied rewrites 97.9%

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

5. Final simplification 97.9%

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

Alternative 8: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1 \cdot v}{u \cdot u}\\ \mathbf{if}\;u \leq -7.4 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5.3 \cdot 10^{+129}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* t1 v) (* u u))))
   (if (<= u -7.4e+76) t_1 (if (<= u 5.3e+129) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 * v) / (u * u);
	double tmp;
	if (u <= -7.4e+76) {
		tmp = t_1;
	} else if (u <= 5.3e+129) {
		tmp = -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 = (t1 * v) / (u * u)
    if (u <= (-7.4d+76)) then
        tmp = t_1
    else if (u <= 5.3d+129) then
        tmp = -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 = (t1 * v) / (u * u);
	double tmp;
	if (u <= -7.4e+76) {
		tmp = t_1;
	} else if (u <= 5.3e+129) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 * v) / (u * u)
	tmp = 0
	if u <= -7.4e+76:
		tmp = t_1
	elif u <= 5.3e+129:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 * v) / Float64(u * u))
	tmp = 0.0
	if (u <= -7.4e+76)
		tmp = t_1;
	elseif (u <= 5.3e+129)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 * v) / (u * u);
	tmp = 0.0;
	if (u <= -7.4e+76)
		tmp = t_1;
	elseif (u <= 5.3e+129)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -7.4e+76], t$95$1, If[LessEqual[u, 5.3e+129], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -7.3999999999999999e76 or 5.2999999999999999e129 < u

   1. Initial program 81.3%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 79.1

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

   7. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. neg-mul-1 N/A

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

   9. Applied rewrites 72.6%

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

   if -7.3999999999999999e76 < u < 5.2999999999999999e129

   1. Initial program 73.3%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.7

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

   5. Applied rewrites 67.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.4 \cdot 10^{+76}:\\ \;\;\;\;\frac{t1 \cdot v}{u \cdot u}\\ \mathbf{elif}\;u \leq 5.3 \cdot 10^{+129}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-frac-neg2 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-neg.f64 99.0

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.0

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

4. Applied rewrites 99.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lift-neg.f64 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 98.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 98.8

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 98.8

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

6. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. neg-mul-1 N/A

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

   8. lift-neg.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*r/ N/A

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

   12. lower-/.f64 N/A

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

8. Applied rewrites 97.9%

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

10. Applied rewrites 94.9%

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

11. Final simplification 94.9%

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

Alternative 10: 53.4% accurate, 2.1× 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 76.0%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 52.2

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

5. Applied rewrites 52.2%

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

Alternative 11: 13.6% accurate, 2.5× 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 76.0%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 52.2

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

5. Applied rewrites 52.2%

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

7. Applied rewrites 51.9%

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

9. Applied rewrites 11.5%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))