Rosa's DopplerBench

Percentage Accurate: 73.1% → 97.7%

Time: 11.1s

Alternatives: 8

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: 73.1% 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: 97.7% 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 79.4%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. 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 98.8

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

4. Applied egg-rr 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{\frac{1}{v} \cdot \mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.6e+153)
   (/ -1.0 (* (/ 1.0 v) (fma u 2.0 t1)))
   (if (<= t1 4.6e+64)
     (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))
     (- (/ v (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.6e+153) {
		tmp = -1.0 / ((1.0 / v) * fma(u, 2.0, t1));
	} else if (t1 <= 4.6e+64) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = -(v / (t1 + u));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.6e+153)
		tmp = Float64(-1.0 / Float64(Float64(1.0 / v) * fma(u, 2.0, t1)));
	elseif (t1 <= 4.6e+64)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = Float64(-Float64(v / Float64(t1 + u)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.6e+153], N[(-1.0 / N[(N[(1.0 / v), $MachinePrecision] * N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.6e+64], N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.5999999999999997e153

   1. Initial program 54.6%

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

      1. *-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 99.9

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

   4. Applied egg-rr 99.9%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

      8. div-inv N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. clear-num N/A

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

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

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

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

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

      14. /-lowering-/.f64 N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 63.5

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

   6. Applied egg-rr 63.5%

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

   7. Taylor expanded in u around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. distribute-lft-out N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 91.7

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

   9. Simplified 91.7%

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

   if -5.5999999999999997e153 < t1 < 4.6e64

   1. Initial program 87.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 89.9

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

   4. Applied egg-rr 89.9%

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

   if 4.6e64 < t1

   1. Initial program 61.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 100.0

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

   4. Applied egg-rr 100.0%

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

   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 96.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{\frac{1}{v} \cdot \mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.5e+153)
   (/ v (- t1))
   (if (<= t1 4.6e+64)
     (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))
     (- (/ v (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.5e+153) {
		tmp = v / -t1;
	} else if (t1 <= 4.6e+64) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = -(v / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-5.5d+153)) then
        tmp = v / -t1
    else if (t1 <= 4.6d+64) then
        tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
    else
        tmp = -(v / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.5e+153) {
		tmp = v / -t1;
	} else if (t1 <= 4.6e+64) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = -(v / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.5e+153:
		tmp = v / -t1
	elif t1 <= 4.6e+64:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = -(v / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.5e+153)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 4.6e+64)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = Float64(-Float64(v / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5.5e+153)
		tmp = v / -t1;
	elseif (t1 <= 4.6e+64)
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	else
		tmp = -(v / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.5e+153], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 4.6e+64], N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.5000000000000003e153

   1. Initial program 54.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 90.9

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

   5. Simplified 90.9%

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

   if -5.5000000000000003e153 < t1 < 4.6e64

   1. Initial program 87.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 89.9

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

   4. Applied egg-rr 89.9%

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

   if 4.6e64 < t1

   1. Initial program 61.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 100.0

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

   4. Applied egg-rr 100.0%

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

   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 96.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -4.6 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;t1 \cdot \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (+ t1 u)))))
   (if (<= t1 -4.6e+62)
     t_1
     (if (<= t1 4.6e+64) (* t1 (/ (- v) (* (+ t1 u) (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / (t1 + u));
	double tmp;
	if (t1 <= -4.6e+62) {
		tmp = t_1;
	} else if (t1 <= 4.6e+64) {
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -(v / (t1 + u))
    if (t1 <= (-4.6d+62)) then
        tmp = t_1
    else if (t1 <= 4.6d+64) then
        tmp = t1 * (-v / ((t1 + u) * (t1 + u)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -(v / (t1 + u));
	double tmp;
	if (t1 <= -4.6e+62) {
		tmp = t_1;
	} else if (t1 <= 4.6e+64) {
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / (t1 + u))
	tmp = 0
	if t1 <= -4.6e+62:
		tmp = t_1
	elif t1 <= 4.6e+64:
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-Float64(v / Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -4.6e+62)
		tmp = t_1;
	elseif (t1 <= 4.6e+64)
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(v / (t1 + u));
	tmp = 0.0;
	if (t1 <= -4.6e+62)
		tmp = t_1;
	elseif (t1 <= 4.6e+64)
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = (-N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t1, -4.6e+62], t$95$1, If[LessEqual[t1, 4.6e+64], N[(t1 * N[((-v) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.59999999999999968e62 or 4.6e64 < t1

   1. Initial program 63.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 100.0

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

   4. Applied egg-rr 100.0%

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

   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 90.5%

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

   if -4.59999999999999968e62 < t1 < 4.6e64

   1. Initial program 88.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. neg-lowering-neg.f64 89.6

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

   4. Applied egg-rr 89.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.6 \cdot 10^{+62}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{+64}:\\ \;\;\;\;t1 \cdot \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.9 \cdot 10^{-126}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v (+ t1 u)))))
   (if (<= t1 -1.3e-14)
     t_1
     (if (<= t1 4.9e-126) (* (- 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 <= -1.3e-14) {
		tmp = t_1;
	} else if (t1 <= 4.9e-126) {
		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 <= (-1.3d-14)) then
        tmp = t_1
    else if (t1 <= 4.9d-126) 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 <= -1.3e-14) {
		tmp = t_1;
	} else if (t1 <= 4.9e-126) {
		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 <= -1.3e-14:
		tmp = t_1
	elif t1 <= 4.9e-126:
		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 <= -1.3e-14)
		tmp = t_1;
	elseif (t1 <= 4.9e-126)
		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 <= -1.3e-14)
		tmp = t_1;
	elseif (t1 <= 4.9e-126)
		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, -1.3e-14], t$95$1, If[LessEqual[t1, 4.9e-126], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.29999999999999998e-14 or 4.9000000000000001e-126 < t1

   1. Initial program 75.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.3

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

   4. Applied egg-rr 99.3%

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

   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 80.0%

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

   if -1.29999999999999998e-14 < t1 < 4.9000000000000001e-126

   1. Initial program 86.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 84.1

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

   5. Simplified 84.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-14}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.9 \cdot 10^{-126}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{v}{u}\\ \mathbf{if}\;u \leq -7 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (/ v u))))
   (if (<= u -7e+154) t_1 (if (<= u 5.1e+103) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -(v / u);
	double tmp;
	if (u <= -7e+154) {
		tmp = t_1;
	} else if (u <= 5.1e+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 = -(v / u)
    if (u <= (-7d+154)) then
        tmp = t_1
    else if (u <= 5.1d+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 = -(v / u);
	double tmp;
	if (u <= -7e+154) {
		tmp = t_1;
	} else if (u <= 5.1e+103) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -(v / u)
	tmp = 0
	if u <= -7e+154:
		tmp = t_1
	elif u <= 5.1e+103:
		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 <= -7e+154)
		tmp = t_1;
	elseif (u <= 5.1e+103)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -(v / u);
	tmp = 0.0;
	if (u <= -7e+154)
		tmp = t_1;
	elseif (u <= 5.1e+103)
		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, -7e+154], t$95$1, If[LessEqual[u, 5.1e+103], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -7.00000000000000041e154 or 5.1000000000000002e103 < u

   1. Initial program 85.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 50.2%

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

   8. Taylor expanded in t1 around 0

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

   10. Simplified 46.4%

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

   if -7.00000000000000041e154 < u < 5.1000000000000002e103

   1. Initial program 77.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. /-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 67.0

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

   5. Simplified 67.0%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7 \cdot 10^{+154}:\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{elif}\;u \leq 5.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.2% 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 79.4%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. 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 98.0

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

4. Applied egg-rr 98.0%

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

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 62.4%

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

8. Final simplification 62.4%

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

Alternative 8: 17.1% 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 79.4%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. 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 98.0

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

4. Applied egg-rr 98.0%

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

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 62.4%

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

8. Taylor expanded in t1 around 0

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

10. Simplified 19.8%

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

11. Final simplification 19.8%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))