Rosa's DopplerBench

Percentage Accurate: 72.7% → 98.2%

Time: 11.1s

Alternatives: 11

Speedup: 0.8×

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: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

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

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)) * 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 72.8%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 98.5

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 5 \cdot 10^{+146}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -9.5e+139)
   (/ v (- (- t1) u))
   (if (<= t1 5e+146) (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))) (/ v (- t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9.5e+139) {
		tmp = v / (-t1 - u);
	} else if (t1 <= 5e+146) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + 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 <= (-9.5d+139)) then
        tmp = v / (-t1 - u)
    else if (t1 <= 5d+146) then
        tmp = v * (-t1 / ((t1 + u) * (t1 + 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 <= -9.5e+139) {
		tmp = v / (-t1 - u);
	} else if (t1 <= 5e+146) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -9.5e+139:
		tmp = v / (-t1 - u)
	elif t1 <= 5e+146:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -9.5e+139)
		tmp = Float64(v / Float64(Float64(-t1) - u));
	elseif (t1 <= 5e+146)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -9.5e+139)
		tmp = v / (-t1 - u);
	elseif (t1 <= 5e+146)
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -9.5e+139], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5e+146], N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -9.5000000000000002e139

   1. Initial program 56.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t1 around inf

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

   7. Applied rewrites 92.9%

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

   if -9.5000000000000002e139 < t1 < 4.9999999999999999e146

   1. Initial program 83.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 85.8

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

   4. Applied rewrites 85.8%

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

   if 4.9999999999999999e146 < t1

   1. Initial program 29.7%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 84.9

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

   5. Applied rewrites 84.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 5 \cdot 10^{+146}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) - u\\ t_2 := \frac{v}{t\_1}\\ \mathbf{if}\;t1 \leq -1.26 \cdot 10^{+132}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 8 \cdot 10^{+130}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- t1) u)) (t_2 (/ v t_1)))
   (if (<= t1 -1.26e+132)
     t_2
     (if (<= t1 8e+130) (* t1 (/ v (* (+ t1 u) t_1))) t_2))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double t_2 = v / t_1;
	double tmp;
	if (t1 <= -1.26e+132) {
		tmp = t_2;
	} else if (t1 <= 8e+130) {
		tmp = t1 * (v / ((t1 + u) * t_1));
	} else {
		tmp = t_2;
	}
	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 = -t1 - u
    t_2 = v / t_1
    if (t1 <= (-1.26d+132)) then
        tmp = t_2
    else if (t1 <= 8d+130) then
        tmp = t1 * (v / ((t1 + u) * t_1))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double t_2 = v / t_1;
	double tmp;
	if (t1 <= -1.26e+132) {
		tmp = t_2;
	} else if (t1 <= 8e+130) {
		tmp = t1 * (v / ((t1 + u) * t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 - u
	t_2 = v / t_1
	tmp = 0
	if t1 <= -1.26e+132:
		tmp = t_2
	elif t1 <= 8e+130:
		tmp = t1 * (v / ((t1 + u) * t_1))
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) - u)
	t_2 = Float64(v / t_1)
	tmp = 0.0
	if (t1 <= -1.26e+132)
		tmp = t_2;
	elseif (t1 <= 8e+130)
		tmp = Float64(t1 * Float64(v / Float64(Float64(t1 + u) * t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 - u;
	t_2 = v / t_1;
	tmp = 0.0;
	if (t1 <= -1.26e+132)
		tmp = t_2;
	elseif (t1 <= 8e+130)
		tmp = t1 * (v / ((t1 + u) * t_1));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) - u), $MachinePrecision]}, Block[{t$95$2 = N[(v / t$95$1), $MachinePrecision]}, If[LessEqual[t1, -1.26e+132], t$95$2, If[LessEqual[t1, 8e+130], N[(t1 * N[(v / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.25999999999999999e132 or 8.0000000000000005e130 < t1

   1. Initial program 47.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t1 around inf

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

   7. Applied rewrites 88.9%

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

   if -1.25999999999999999e132 < t1 < 8.0000000000000005e130

   1. Initial program 83.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.8

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

   4. Applied rewrites 83.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.26 \cdot 10^{+132}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 8 \cdot 10^{+130}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u}\\ \mathbf{if}\;t1 \leq -7.2 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{-79}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) u))))
   (if (<= t1 -7.2e-29)
     t_1
     (if (<= t1 6.2e-79) (* (- 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 <= -7.2e-29) {
		tmp = t_1;
	} else if (t1 <= 6.2e-79) {
		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 <= (-7.2d-29)) then
        tmp = t_1
    else if (t1 <= 6.2d-79) 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 <= -7.2e-29) {
		tmp = t_1;
	} else if (t1 <= 6.2e-79) {
		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 <= -7.2e-29:
		tmp = t_1
	elif t1 <= 6.2e-79:
		tmp = -t1 * (v / (u * u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - u))
	tmp = 0.0
	if (t1 <= -7.2e-29)
		tmp = t_1;
	elseif (t1 <= 6.2e-79)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(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 <= -7.2e-29)
		tmp = t_1;
	elseif (t1 <= 6.2e-79)
		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, -7.2e-29], t$95$1, If[LessEqual[t1, 6.2e-79], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.19999999999999948e-29 or 6.1999999999999999e-79 < t1

   1. Initial program 66.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Applied rewrites 78.9%

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

   if -7.19999999999999948e-29 < t1 < 6.1999999999999999e-79

   1. Initial program 80.8%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 77.2

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

   5. Applied rewrites 77.2%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{-79}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{if}\;u \leq -2.2 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 0.0075:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- t1) (/ v (* u u)))))
   (if (<= u -2.2e+37) t_1 (if (<= u 0.0075) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 * (v / (u * u));
	double tmp;
	if (u <= -2.2e+37) {
		tmp = t_1;
	} else if (u <= 0.0075) {
		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 <= (-2.2d+37)) then
        tmp = t_1
    else if (u <= 0.0075d0) 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 <= -2.2e+37) {
		tmp = t_1;
	} else if (u <= 0.0075) {
		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 <= -2.2e+37:
		tmp = t_1
	elif u <= 0.0075:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) * Float64(v / Float64(u * u)))
	tmp = 0.0
	if (u <= -2.2e+37)
		tmp = t_1;
	elseif (u <= 0.0075)
		tmp = Float64(v / Float64(-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 <= -2.2e+37)
		tmp = t_1;
	elseif (u <= 0.0075)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.2e+37], t$95$1, If[LessEqual[u, 0.0075], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.2000000000000001e37 or 0.0074999999999999997 < u

   1. Initial program 79.3%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -2.2000000000000001e37 < u < 0.0074999999999999997

   1. Initial program 67.5%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+37}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 0.0075:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   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. lower-neg.f64 97.6

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

4. Applied rewrites 97.6%

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

5. Final simplification 97.6%

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

Alternative 7: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(u, v, t1)
	return Float64(Float64(t1 / Float64(t1 + u)) * Float64(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[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.8%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 97.3

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

4. Applied rewrites 97.3%

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

5. Final simplification 97.3%

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

Alternative 8: 58.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.1 \cdot 10^{+165}:\\ \;\;\;\;\frac{v}{u \cdot -2}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.1e+165)
   (/ v (* u -2.0))
   (if (<= u 3.4e+84) (/ v (- t1)) (/ -0.5 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.1e+165) {
		tmp = v / (u * -2.0);
	} else if (u <= 3.4e+84) {
		tmp = v / -t1;
	} else {
		tmp = -0.5 / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.1d+165)) then
        tmp = v / (u * (-2.0d0))
    else if (u <= 3.4d+84) then
        tmp = v / -t1
    else
        tmp = (-0.5d0) / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.1e+165) {
		tmp = v / (u * -2.0);
	} else if (u <= 3.4e+84) {
		tmp = v / -t1;
	} else {
		tmp = -0.5 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.1e+165:
		tmp = v / (u * -2.0)
	elif u <= 3.4e+84:
		tmp = v / -t1
	else:
		tmp = -0.5 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.1e+165)
		tmp = Float64(v / Float64(u * -2.0));
	elseif (u <= 3.4e+84)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(-0.5 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.1e+165)
		tmp = v / (u * -2.0);
	elseif (u <= 3.4e+84)
		tmp = v / -t1;
	else
		tmp = -0.5 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.1e+165], N[(v / N[(u * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.4e+84], N[(v / (-t1)), $MachinePrecision], N[(-0.5 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.1000000000000004e165

   1. Initial program 64.5%

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

   3. Taylor expanded in t1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 47.2

         \[\leadsto \frac{\color{blue}{v \cdot -0.5}}{u} \]

   8. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\frac{v \cdot -0.5}{u}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 50.5

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

   10. Applied rewrites 50.5%

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

   if -5.1000000000000004e165 < u < 3.3999999999999998e84

   1. Initial program 71.0%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

   if 3.3999999999999998e84 < u

   1. Initial program 82.1%

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

   3. Taylor expanded in t1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 40.1

         \[\leadsto \frac{\color{blue}{v \cdot -0.5}}{u} \]

   8. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{v \cdot -0.5}{u}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 40.7

         \[\leadsto \frac{-0.5}{\color{blue}{\frac{u}{v}}} \]

   10. Applied rewrites 40.7%

       \[\leadsto \color{blue}{\frac{-0.5}{\frac{u}{v}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.1 \cdot 10^{+165}:\\ \;\;\;\;\frac{v}{u \cdot -2}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2}\\ \mathbf{if}\;u \leq -5.1 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (* u -2.0))))
   (if (<= u -5.1e+165) t_1 (if (<= u 3.4e+84) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (u * -2.0);
	double tmp;
	if (u <= -5.1e+165) {
		tmp = t_1;
	} else if (u <= 3.4e+84) {
		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 = v / (u * (-2.0d0))
    if (u <= (-5.1d+165)) then
        tmp = t_1
    else if (u <= 3.4d+84) 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 = v / (u * -2.0);
	double tmp;
	if (u <= -5.1e+165) {
		tmp = t_1;
	} else if (u <= 3.4e+84) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (u * -2.0)
	tmp = 0
	if u <= -5.1e+165:
		tmp = t_1
	elif u <= 3.4e+84:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(u * -2.0))
	tmp = 0.0
	if (u <= -5.1e+165)
		tmp = t_1;
	elseif (u <= 3.4e+84)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (u * -2.0);
	tmp = 0.0;
	if (u <= -5.1e+165)
		tmp = t_1;
	elseif (u <= 3.4e+84)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(u * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -5.1e+165], t$95$1, If[LessEqual[u, 3.4e+84], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -5.1000000000000004e165 or 3.3999999999999998e84 < u

   1. Initial program 76.5%

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

   3. Taylor expanded in t1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 42.3

         \[\leadsto \frac{\color{blue}{v \cdot -0.5}}{u} \]

   8. Applied rewrites 42.3%

      \[\leadsto \color{blue}{\frac{v \cdot -0.5}{u}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 43.3

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

   10. Applied rewrites 43.3%

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

   if -5.1000000000000004e165 < u < 3.3999999999999998e84

   1. Initial program 71.0%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.1 \cdot 10^{+165}:\\ \;\;\;\;\frac{v}{u \cdot -2}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-0.5}{u}\\ \mathbf{if}\;u \leq -5.1 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ -0.5 u))))
   (if (<= u -5.1e+165) t_1 (if (<= u 3.4e+84) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-0.5 / u);
	double tmp;
	if (u <= -5.1e+165) {
		tmp = t_1;
	} else if (u <= 3.4e+84) {
		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 = v * ((-0.5d0) / u)
    if (u <= (-5.1d+165)) then
        tmp = t_1
    else if (u <= 3.4d+84) 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 = v * (-0.5 / u);
	double tmp;
	if (u <= -5.1e+165) {
		tmp = t_1;
	} else if (u <= 3.4e+84) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-0.5 / u)
	tmp = 0
	if u <= -5.1e+165:
		tmp = t_1
	elif u <= 3.4e+84:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(-0.5 / u))
	tmp = 0.0
	if (u <= -5.1e+165)
		tmp = t_1;
	elseif (u <= 3.4e+84)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-0.5 / u);
	tmp = 0.0;
	if (u <= -5.1e+165)
		tmp = t_1;
	elseif (u <= 3.4e+84)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(-0.5 / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -5.1e+165], t$95$1, If[LessEqual[u, 3.4e+84], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -5.1000000000000004e165 or 3.3999999999999998e84 < u

   1. Initial program 76.5%

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

   3. Taylor expanded in t1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 42.3

         \[\leadsto \frac{\color{blue}{v \cdot -0.5}}{u} \]

   8. Applied rewrites 42.3%

      \[\leadsto \color{blue}{\frac{v \cdot -0.5}{u}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 42.3

         \[\leadsto \color{blue}{\frac{-0.5}{u}} \cdot v \]

   10. Applied rewrites 42.3%

       \[\leadsto \color{blue}{\frac{-0.5}{u} \cdot v} \]

   if -5.1000000000000004e165 < u < 3.3999999999999998e84

   1. Initial program 71.0%

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

   3. Taylor expanded in t1 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.1 \cdot 10^{+165}:\\ \;\;\;\;v \cdot \frac{-0.5}{u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-0.5}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.8% 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(v / Float64(-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 72.8%

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

3. Taylor expanded in t1 around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
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 53.5

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

5. Applied rewrites 53.5%

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

6. Final simplification 53.5%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))