Rosa's DopplerBench

Average Error: 18.0 → 0.6

Time: 16.8s

Precision: binary64

Cost: 1032

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot \frac{v}{t1 + u}}{t1 + u}\\ \mathbf{if}\;u \leq -2.4 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 2 \cdot 10^{-47}:\\ \;\;\;\;-v \cdot \frac{\frac{t1}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) (/ v (+ t1 u))) (+ t1 u))))
   (if (<= u -2.4e-124)
     t_1
     (if (<= u 2e-47) (- (* v (/ (/ t1 (+ t1 u)) (+ t1 u)))) t_1))))

C

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

↓

double code(double u, double v, double t1) {
	double t_1 = (-t1 * (v / (t1 + u))) / (t1 + u);
	double tmp;
	if (u <= -2.4e-124) {
		tmp = t_1;
	} else if (u <= 2e-47) {
		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
    code = (-t1 * v) / ((t1 + u) * (t1 + u))
end function

↓

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-t1 * (v / (t1 + u))) / (t1 + u)
    if (u <= (-2.4d-124)) then
        tmp = t_1
    else if (u <= 2d-47) 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) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

↓

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 * (v / (t1 + u))) / (t1 + u);
	double tmp;
	if (u <= -2.4e-124) {
		tmp = t_1;
	} else if (u <= 2e-47) {
		tmp = -(v * ((t1 / (t1 + u)) / (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(u, v, t1):
	t_1 = (-t1 * (v / (t1 + u))) / (t1 + u)
	tmp = 0
	if u <= -2.4e-124:
		tmp = t_1
	elif u <= 2e-47:
		tmp = -(v * ((t1 / (t1 + u)) / (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * Float64(v / Float64(t1 + u))) / Float64(t1 + u))
	tmp = 0.0
	if (u <= -2.4e-124)
		tmp = t_1;
	elseif (u <= 2e-47)
		tmp = Float64(-Float64(v * Float64(Float64(t1 / Float64(t1 + u)) / Float64(t1 + u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 * (v / (t1 + u))) / (t1 + u);
	tmp = 0.0;
	if (u <= -2.4e-124)
		tmp = t_1;
	elseif (u <= 2e-47)
		tmp = -(v * ((t1 / (t1 + u)) / (t1 + u)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.4e-124], t$95$1, If[LessEqual[u, 2e-47], (-N[(v * N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.39999999999999992e-124 or 1.9999999999999999e-47 < u

   1. Initial program 14.6

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

   2. Simplified 1.3

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

      [Start] 14.6	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      rational.json-simplify-46 [=>] 8.8	\[ \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]
      rational.json-simplify-49 [=>] 1.3	\[ \frac{\color{blue}{v \cdot \frac{-t1}{t1 + u}}}{t1 + u} \]
      rational.json-simplify-49 [=>] 1.3	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   3. Applied egg-rr 0.7

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

   if -2.39999999999999992e-124 < u < 1.9999999999999999e-47

   1. Initial program 24.8

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

   2. Simplified 1.5

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

      [Start] 24.8	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      rational.json-simplify-46 [=>] 16.8	\[ \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]
      rational.json-simplify-49 [=>] 0.9	\[ \frac{\color{blue}{v \cdot \frac{-t1}{t1 + u}}}{t1 + u} \]
      rational.json-simplify-49 [=>] 1.5	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   3. Applied egg-rr 0.3

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{-124}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{t1 + u}}{t1 + u}\\ \mathbf{elif}\;u \leq 2 \cdot 10^{-47}:\\ \;\;\;\;-v \cdot \frac{\frac{t1}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{t1 + u}}{t1 + u}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.5
Cost	1300

\[\begin{array}{l} \mathbf{if}\;u \leq -9.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{t1 + u}{v}}}{u}\\ \mathbf{elif}\;u \leq 4.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{t1}{u}}{t1 + u} \cdot \left(-v\right)\\ \mathbf{elif}\;u \leq 7000000:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;u \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u + t1}}{\frac{u}{v}}\\ \end{array} \]

Alternative 2
Error	15.8
Cost	1040

\[\begin{array}{l} \mathbf{if}\;u \leq -2.05 \cdot 10^{+39}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq 7.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq 1.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 \cdot \left(v \cdot \frac{t1}{u}\right)}{u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{-8}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \end{array} \]

Alternative 3
Error	15.7
Cost	1040

\[\begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq 4.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{t1}{u}}{t1 + u} \cdot \left(-v\right)\\ \mathbf{elif}\;u \leq 3 \cdot 10^{-9}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \end{array} \]

Alternative 4
Error	13.9
Cost	776

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -4.4 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-57}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.9
Cost	776

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.95 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 5.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	3.5
Cost	768

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

Alternative 7
Error	1.4
Cost	768

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

Alternative 8
Error	28.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{+105}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 1.02 \cdot 10^{+108}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \end{array} \]

Alternative 9
Error	28.0
Cost	520

\[\begin{array}{l} t_1 := \frac{-v}{u}\\ \mathbf{if}\;u \leq -3.2 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+155}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	25.4
Cost	384

\[\frac{-v}{t1 + u} \]

Alternative 11
Error	31.0
Cost	256

\[-\frac{v}{t1} \]

Alternative 12
Error	54.9
Cost	192

\[\frac{v}{t1} \]

Reproduce

herbie shell --seed 2023075 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))