Rosa's DopplerBench

Percentage Accurate: 72.7% → 98.3%

Time: 10.7s

Alternatives: 9

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: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. *-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. neg-mul-1 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. neg-lowering-neg.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. +-lowering-+.f64 98.5

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t1 + u\right) \cdot \left(t1 + u\right)\\ \mathbf{if}\;\frac{v \cdot \left(-t1\right)}{t\_1} \leq 4 \cdot 10^{+228}:\\ \;\;\;\;v \cdot \frac{-t1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (+ t1 u) (+ t1 u))))
   (if (<= (/ (* v (- t1)) t_1) 4e+228) (* 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) <= 4e+228) {
		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) <= 4d+228) 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) <= 4e+228) {
		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) <= 4e+228:
		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) <= 4e+228)
		tmp = Float64(v * Float64(Float64(-t1) / t_1));
	else
		tmp = Float64(-Float64(v / 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) <= 4e+228)
		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], 4e+228], 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))) < 3.9999999999999997e228

   1. Initial program 84.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 87.4

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

   4. Applied egg-rr 87.4%

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

   if 3.9999999999999997e228 < (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u)))

   1. Initial program 9.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. /-lowering-/.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. neg-lowering-neg.f64 84.0

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

   5. Simplified 84.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{v \cdot \left(-t1\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \leq 4 \cdot 10^{+228}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-86}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{-48}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -9e-86)
   (- (/ v (+ t1 u)))
   (if (<= t1 7e-48) (* (/ v u) (/ t1 (- u))) (/ -1.0 (/ (+ t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9e-86) {
		tmp = -(v / (t1 + u));
	} else if (t1 <= 7e-48) {
		tmp = (v / u) * (t1 / -u);
	} else {
		tmp = -1.0 / ((t1 + 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 (t1 <= (-9d-86)) then
        tmp = -(v / (t1 + u))
    else if (t1 <= 7d-48) then
        tmp = (v / u) * (t1 / -u)
    else
        tmp = (-1.0d0) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9e-86) {
		tmp = -(v / (t1 + u));
	} else if (t1 <= 7e-48) {
		tmp = (v / u) * (t1 / -u);
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -9e-86:
		tmp = -(v / (t1 + u))
	elif t1 <= 7e-48:
		tmp = (v / u) * (t1 / -u)
	else:
		tmp = -1.0 / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -9e-86)
		tmp = Float64(-Float64(v / Float64(t1 + u)));
	elseif (t1 <= 7e-48)
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	else
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -9e-86)
		tmp = -(v / (t1 + u));
	elseif (t1 <= 7e-48)
		tmp = (v / u) * (t1 / -u);
	else
		tmp = -1.0 / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -9e-86], (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t1, 7e-48], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -8.9999999999999995e-86

   1. Initial program 63.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 79.9%

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

   if -8.9999999999999995e-86 < t1 < 6.99999999999999982e-48

   1. Initial program 77.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 80.2

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

   4. Applied egg-rr 80.2%

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

   5. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 74.0

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

   7. Simplified 74.0%

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

      1. associate-*l/ N/A

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

      2. frac-2neg N/A

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

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

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

      4. *-commutative N/A

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

      5. remove-double-neg N/A

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

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

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

      7. times-frac N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. /-lowering-/.f64 80.1

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

   9. Applied egg-rr 80.1%

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

   if 6.99999999999999982e-48 < t1

   1. Initial program 70.5%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 86.7%

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

      1. distribute-frac-neg2 N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-lowering-+.f64 86.8

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

   9. Applied egg-rr 86.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-86}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{-48}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-139}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{-47}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5e-139)
   (- (/ v (+ t1 u)))
   (if (<= t1 1.6e-47) (* (- v) (/ t1 (* u u))) (/ -1.0 (/ (+ t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5e-139) {
		tmp = -(v / (t1 + u));
	} else if (t1 <= 1.6e-47) {
		tmp = -v * (t1 / (u * u));
	} else {
		tmp = -1.0 / ((t1 + 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 (t1 <= (-5d-139)) then
        tmp = -(v / (t1 + u))
    else if (t1 <= 1.6d-47) then
        tmp = -v * (t1 / (u * u))
    else
        tmp = (-1.0d0) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5e-139) {
		tmp = -(v / (t1 + u));
	} else if (t1 <= 1.6e-47) {
		tmp = -v * (t1 / (u * u));
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5e-139:
		tmp = -(v / (t1 + u))
	elif t1 <= 1.6e-47:
		tmp = -v * (t1 / (u * u))
	else:
		tmp = -1.0 / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5e-139)
		tmp = Float64(-Float64(v / Float64(t1 + u)));
	elseif (t1 <= 1.6e-47)
		tmp = Float64(Float64(-v) * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5e-139)
		tmp = -(v / (t1 + u));
	elseif (t1 <= 1.6e-47)
		tmp = -v * (t1 / (u * u));
	else
		tmp = -1.0 / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5e-139], (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t1, 1.6e-47], N[((-v) * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.00000000000000034e-139

   1. Initial program 65.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 78.3%

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

   if -5.00000000000000034e-139 < t1 < 1.6e-47

   1. Initial program 76.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 78.8

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

   4. Applied egg-rr 78.8%

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

   5. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 75.2

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

   7. Simplified 75.2%

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

   if 1.6e-47 < t1

   1. Initial program 70.5%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 86.7%

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

      1. distribute-frac-neg2 N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-lowering-+.f64 86.8

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

   9. Applied egg-rr 86.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-139}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{-47}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.7 \cdot 10^{-144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.4 \cdot 10^{-47}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (+ t1 u)))))
   (if (<= t1 -3.7e-144)
     t_1
     (if (<= t1 1.4e-47) (* (- v) (/ t1 (* u u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / (t1 + u));
	double tmp;
	if (t1 <= -3.7e-144) {
		tmp = t_1;
	} else if (t1 <= 1.4e-47) {
		tmp = -v * (t1 / (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 <= (-3.7d-144)) then
        tmp = t_1
    else if (t1 <= 1.4d-47) then
        tmp = -v * (t1 / (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 <= -3.7e-144) {
		tmp = t_1;
	} else if (t1 <= 1.4e-47) {
		tmp = -v * (t1 / (u * u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / (t1 + u))
	tmp = 0
	if t1 <= -3.7e-144:
		tmp = t_1
	elif t1 <= 1.4e-47:
		tmp = -v * (t1 / (u * u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -3.7e-144)
		tmp = t_1;
	elseif (t1 <= 1.4e-47)
		tmp = Float64(Float64(-v) * Float64(t1 / 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 <= -3.7e-144)
		tmp = t_1;
	elseif (t1 <= 1.4e-47)
		tmp = -v * (t1 / (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, -3.7e-144], t$95$1, If[LessEqual[t1, 1.4e-47], N[((-v) * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.7000000000000003e-144 or 1.39999999999999996e-47 < t1

   1. Initial program 67.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 81.5%

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

   if -3.7000000000000003e-144 < t1 < 1.39999999999999996e-47

   1. Initial program 76.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 78.8

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

   4. Applied egg-rr 78.8%

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

   5. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 75.2

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

   7. Simplified 75.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.7 \cdot 10^{-144}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.4 \cdot 10^{-47}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.05 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.8 \cdot 10^{-47}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (+ t1 u)))))
   (if (<= t1 -1.05e-85)
     t_1
     (if (<= t1 2.8e-47) (* (- 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 <= -1.05e-85) {
		tmp = t_1;
	} else if (t1 <= 2.8e-47) {
		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 <= (-1.05d-85)) then
        tmp = t_1
    else if (t1 <= 2.8d-47) 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 <= -1.05e-85) {
		tmp = t_1;
	} else if (t1 <= 2.8e-47) {
		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 <= -1.05e-85:
		tmp = t_1
	elif t1 <= 2.8e-47:
		tmp = -t1 * (v / (u * u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -1.05e-85)
		tmp = t_1;
	elseif (t1 <= 2.8e-47)
		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 <= -1.05e-85)
		tmp = t_1;
	elseif (t1 <= 2.8e-47)
		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, -1.05e-85], t$95$1, If[LessEqual[t1, 2.8e-47], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.05e-85 or 2.79999999999999993e-47 < t1

   1. Initial program 66.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 82.6%

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

   if -1.05e-85 < t1 < 2.79999999999999993e-47

   1. Initial program 77.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.f64 N/A

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

      14. neg-lowering-neg.f64 74.0

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

   5. Simplified 74.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{-85}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.8 \cdot 10^{-47}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{u}\\ \mathbf{if}\;u \leq -1.36 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+144}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v u))))
   (if (<= u -1.36e+138) t_1 (if (<= u 5.2e+144) (- (/ v t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / u);
	double tmp;
	if (u <= -1.36e+138) {
		tmp = t_1;
	} else if (u <= 5.2e+144) {
		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)
    if (u <= (-1.36d+138)) then
        tmp = t_1
    else if (u <= 5.2d+144) 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);
	double tmp;
	if (u <= -1.36e+138) {
		tmp = t_1;
	} else if (u <= 5.2e+144) {
		tmp = -(v / t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / u)
	tmp = 0
	if u <= -1.36e+138:
		tmp = t_1
	elif u <= 5.2e+144:
		tmp = -(v / t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / u))
	tmp = 0.0
	if (u <= -1.36e+138)
		tmp = t_1;
	elseif (u <= 5.2e+144)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(v / u);
	tmp = 0.0;
	if (u <= -1.36e+138)
		tmp = t_1;
	elseif (u <= 5.2e+144)
		tmp = -(v / t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / u), $MachinePrecision])}, If[LessEqual[u, -1.36e+138], t$95$1, If[LessEqual[u, 5.2e+144], (-N[(v / t1), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.35999999999999995e138 or 5.1999999999999998e144 < u

   1. Initial program 79.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Simplified 44.7%

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

   8. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 39.4

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

   10. Simplified 39.4%

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

   if -1.35999999999999995e138 < u < 5.1999999999999998e144

   1. Initial program 66.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. /-lowering-/.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. neg-lowering-neg.f64 73.6

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

   5. Simplified 73.6%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.36 \cdot 10^{+138}:\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+144}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. associate-*r/ N/A

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

   6. /-lowering-/.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. neg-lowering-neg.f64 N/A

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

   11. +-lowering-+.f64 98.3

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 65.8%

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

8. Final simplification 65.8%

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

Alternative 9: 16.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. associate-*r/ N/A

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

   6. /-lowering-/.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. neg-lowering-neg.f64 N/A

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

   11. +-lowering-+.f64 98.3

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in t1 around inf

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

7. Simplified 65.8%

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

8. Taylor expanded in t1 around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. /-lowering-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-lowering-neg.f64 17.1

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

10. Simplified 17.1%

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

11. Final simplification 17.1%

    \[\leadsto -\frac{v}{u} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))