Rosa's DopplerBench

Percentage Accurate: 72.7% → 97.9%

Time: 11.2s

Alternatives: 10

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6.8 \cdot 10^{+140}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -2.6e+158)
     t_1
     (if (<= t1 6.8e+140) (* v (/ (- t1) (* (+ t1 u) (+ t1 u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.6e+158) {
		tmp = t_1;
	} else if (t1 <= 6.8e+140) {
		tmp = v * (-t1 / ((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 / ((u * (-2.0d0)) - t1)
    if (t1 <= (-2.6d+158)) then
        tmp = t_1
    else if (t1 <= 6.8d+140) then
        tmp = v * (-t1 / ((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 / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.6e+158) {
		tmp = t_1;
	} else if (t1 <= 6.8e+140) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -2.6e+158:
		tmp = t_1
	elif t1 <= 6.8e+140:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -2.6e+158)
		tmp = t_1;
	elseif (t1 <= 6.8e+140)
		tmp = Float64(v * Float64(Float64(-t1) / 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 / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -2.6e+158)
		tmp = t_1;
	elseif (t1 <= 6.8e+140)
		tmp = v * (-t1 / ((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[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.6e+158], t$95$1, If[LessEqual[t1, 6.8e+140], N[(v * N[((-t1) / 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 < -2.6e158 or 6.8e140 < t1

   1. Initial program 41.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around 0

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.3

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

   9. Applied rewrites 97.3%

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

   10. Taylor expanded in u around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 92.1

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

   12. Applied rewrites 92.1%

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

   if -2.6e158 < t1 < 6.8e140

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 86.9

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

   4. Applied rewrites 86.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+158}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 6.8 \cdot 10^{+140}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))