Rosa's DopplerBench

Percentage Accurate: 72.5% → 97.9%

Time: 10.4s

Alternatives: 10

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: 72.5% 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: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

   \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}} \]
6. Add Preprocessing

Alternative 2: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t1 + u\right) \cdot \left(t1 + u\right)\\ \mathbf{if}\;\frac{v \cdot \left(-t1\right)}{t\_1} \leq 2 \cdot 10^{+229}:\\ \;\;\;\;v \cdot \frac{-t1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (+ t1 u) (+ t1 u))))
   (if (<= (/ (* v (- t1)) t_1) 2e+229) (* v (/ (- t1) t_1)) (/ v (- t1)))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 + u) * (t1 + u);
	double tmp;
	if (((v * -t1) / t_1) <= 2e+229) {
		tmp = v * (-t1 / t_1);
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (t1 + u) * (t1 + u)
    if (((v * -t1) / t_1) <= 2d+229) then
        tmp = v * (-t1 / t_1)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 + u) * (t1 + u);
	double tmp;
	if (((v * -t1) / t_1) <= 2e+229) {
		tmp = v * (-t1 / t_1);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 + u) * (t1 + u)
	tmp = 0
	if ((v * -t1) / t_1) <= 2e+229:
		tmp = v * (-t1 / t_1)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 + u) * Float64(t1 + u))
	tmp = 0.0
	if (Float64(Float64(v * Float64(-t1)) / t_1) <= 2e+229)
		tmp = Float64(v * Float64(Float64(-t1) / t_1));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 + u) * (t1 + u);
	tmp = 0.0;
	if (((v * -t1) / t_1) <= 2e+229)
		tmp = v * (-t1 / t_1);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(v * (-t1)), $MachinePrecision] / t$95$1), $MachinePrecision], 2e+229], N[(v * N[((-t1) / t$95$1), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u))) < 2e229

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.4

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

   4. Applied rewrites 88.4%

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

   if 2e229 < (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u)))

   1. Initial program 20.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{v \cdot \left(-t1\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \leq 2 \cdot 10^{+229}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.7 \cdot 10^{+174}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.7e+174)
   (/ (* v (/ t1 u)) (- (- t1) u))
   (/ v (* (+ t1 u) (- -1.0 (/ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.7e+174) {
		tmp = (v * (t1 / u)) / (-t1 - u);
	} else {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.7d+174)) then
        tmp = (v * (t1 / u)) / (-t1 - u)
    else
        tmp = v / ((t1 + u) * ((-1.0d0) - (u / t1)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.7e+174) {
		tmp = (v * (t1 / u)) / (-t1 - u);
	} else {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.7e+174:
		tmp = (v * (t1 / u)) / (-t1 - u)
	else:
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.7e+174)
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(Float64(-t1) - u));
	else
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(-1.0 - Float64(u / t1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.7e+174)
		tmp = (v * (t1 / u)) / (-t1 - u);
	else
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.7e+174], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[((-t1) - u), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.6999999999999999e174

   1. Initial program 73.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. sqr-neg N/A

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

      7. times-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. associate-*l/ N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in t1 around 0

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

      1. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if -5.6999999999999999e174 < u

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. lower-*.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 95.9

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

   7. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\frac{v}{\left(-1 - \frac{u}{t1}\right) \cdot \left(u + t1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.7 \cdot 10^{+174}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(t1 + u\right) \cdot -1}\\ \mathbf{if}\;t1 \leq -8.5 \cdot 10^{-84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-51}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (* (+ t1 u) -1.0))))
   (if (<= t1 -8.5e-84)
     t_1
     (if (<= t1 2.2e-51) (* (- t1) (/ v (* u u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((t1 + u) * -1.0);
	double tmp;
	if (t1 <= -8.5e-84) {
		tmp = t_1;
	} else if (t1 <= 2.2e-51) {
		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) * (-1.0d0))
    if (t1 <= (-8.5d-84)) then
        tmp = t_1
    else if (t1 <= 2.2d-51) 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) * -1.0);
	double tmp;
	if (t1 <= -8.5e-84) {
		tmp = t_1;
	} else if (t1 <= 2.2e-51) {
		tmp = -t1 * (v / (u * u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((t1 + u) * -1.0)
	tmp = 0
	if t1 <= -8.5e-84:
		tmp = t_1
	elif t1 <= 2.2e-51:
		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) * -1.0))
	tmp = 0.0
	if (t1 <= -8.5e-84)
		tmp = t_1;
	elseif (t1 <= 2.2e-51)
		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) * -1.0);
	tmp = 0.0;
	if (t1 <= -8.5e-84)
		tmp = t_1;
	elseif (t1 <= 2.2e-51)
		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[(N[(t1 + u), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -8.5e-84], t$95$1, If[LessEqual[t1, 2.2e-51], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.4999999999999994e-84 or 2.2e-51 < t1

   1. Initial program 73.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Applied rewrites 86.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 86.5

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

   9. Applied rewrites 86.5%

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

   if -8.4999999999999994e-84 < t1 < 2.2e-51

   1. Initial program 77.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot -1}\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-51}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Derivation

1. Initial program 74.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   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 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 6: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{if}\;u \leq -1.02 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ v (* u u)))))
   (if (<= u -1.02e+109) t_1 (if (<= u 1.6e+103) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (v / (u * u));
	double tmp;
	if (u <= -1.02e+109) {
		tmp = t_1;
	} else if (u <= 1.6e+103) {
		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 <= (-1.02d+109)) then
        tmp = t_1
    else if (u <= 1.6d+103) 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 <= -1.02e+109) {
		tmp = t_1;
	} else if (u <= 1.6e+103) {
		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 <= -1.02e+109:
		tmp = t_1
	elif u <= 1.6e+103:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(v / Float64(u * u)))
	tmp = 0.0
	if (u <= -1.02e+109)
		tmp = t_1;
	elseif (u <= 1.6e+103)
		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 <= -1.02e+109)
		tmp = t_1;
	elseif (u <= 1.6e+103)
		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, -1.02e+109], t$95$1, If[LessEqual[u, 1.6e+103], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.01999999999999994e109 or 1.59999999999999996e103 < u

   1. Initial program 79.7%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 73.5

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

   7. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

      4. neg-mul-1 N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 73.5

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

   9. Applied rewrites 73.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. neg-mul-1 N/A

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

       6. lift-neg.f64 N/A

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

       7. associate-*l/ N/A

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. lift-neg.f64 N/A

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

       11. lift-neg.f64 N/A

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

       12. +-lft-identity N/A

           \[\leadsto \frac{v}{u \cdot u} \cdot \color{blue}{\left(0 + \left(\mathsf{neg}\left(t1\right)\right)\right)} \]

       13. flip-+ N/A

           \[\leadsto \frac{v}{u \cdot u} \cdot \color{blue}{\frac{0 \cdot 0 - \left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right)}{0 - \left(\mathsf{neg}\left(t1\right)\right)}} \]

   11. Applied rewrites 74.1%

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

   if -1.01999999999999994e109 < u < 1.59999999999999996e103

   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 71.6

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

   5. Applied rewrites 71.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{+109}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{t1}{u \cdot u}\\ \mathbf{if}\;u \leq -6.2 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.2 \cdot 10^{+195}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ t1 (* u u)))))
   (if (<= u -6.2e+145) t_1 (if (<= u 1.2e+195) (/ v (* (+ t1 u) -1.0)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (t1 / (u * u));
	double tmp;
	if (u <= -6.2e+145) {
		tmp = t_1;
	} else if (u <= 1.2e+195) {
		tmp = v / ((t1 + u) * -1.0);
	} 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 * u))
    if (u <= (-6.2d+145)) then
        tmp = t_1
    else if (u <= 1.2d+195) then
        tmp = v / ((t1 + u) * (-1.0d0))
    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 * u));
	double tmp;
	if (u <= -6.2e+145) {
		tmp = t_1;
	} else if (u <= 1.2e+195) {
		tmp = v / ((t1 + u) * -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (t1 / (u * u))
	tmp = 0
	if u <= -6.2e+145:
		tmp = t_1
	elif u <= 1.2e+195:
		tmp = v / ((t1 + u) * -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(t1 / Float64(u * u)))
	tmp = 0.0
	if (u <= -6.2e+145)
		tmp = t_1;
	elseif (u <= 1.2e+195)
		tmp = Float64(v / Float64(Float64(t1 + u) * -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (t1 / (u * u));
	tmp = 0.0;
	if (u <= -6.2e+145)
		tmp = t_1;
	elseif (u <= 1.2e+195)
		tmp = v / ((t1 + u) * -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -6.2e+145], t$95$1, If[LessEqual[u, 1.2e+195], N[(v / N[(N[(t1 + u), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -6.19999999999999977e145 or 1.2000000000000001e195 < u

   1. Initial program 81.9%

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

   3. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 82.5

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

   7. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

      4. neg-mul-1 N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 82.5

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

   9. Applied rewrites 82.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

   11. Applied rewrites 80.8%

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

   if -6.19999999999999977e145 < u < 1.2000000000000001e195

   1. Initial program 73.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in t1 around inf

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

   7. Applied rewrites 70.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 70.0

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

   9. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{v}{\left(t1 + u\right) \cdot -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+145}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.2 \cdot 10^{+195}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(t1 + u\right) \cdot -1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (* (+ t1 u) -1.0)))

C

double code(double u, double v, double t1) {
	return v / ((t1 + u) * -1.0);
}

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) * (-1.0d0))
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((t1 + u) * -1.0);
}

Python

def code(u, v, t1):
	return v / ((t1 + u) * -1.0)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(t1 + u) * -1.0))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((t1 + u) * -1.0);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(N[(t1 + u), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in t1 around inf

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

7. Applied rewrites 66.6%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-+.f64 66.6

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

9. Applied rewrites 66.6%

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

Alternative 9: 54.1% 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 74.9%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 57.0

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

5. Applied rewrites 57.0%

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

6. Final simplification 57.0%

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

Alternative 10: 14.5% 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 74.9%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 57.0

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

5. Applied rewrites 57.0%

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

7. Applied rewrites 56.8%

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

9. Applied rewrites 17.6%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))