Rosa's DopplerBench

Average Error: 18.4 → 1.4

Time: 11.5s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

(FPCore (u v t1)
 :precision binary64
 (/ (/ v (+ t1 u)) (+ (- -2.0 (/ u t1)) 1.0)))

C

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

↓

double code(double u, double v, double t1) {
	return (v / (t1 + u)) / ((-2.0 - (u / t1)) + 1.0);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-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
    code = (v / (t1 + u)) / (((-2.0d0) - (u / t1)) + 1.0d0)
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) {
	return (v / (t1 + u)) / ((-2.0 - (u / t1)) + 1.0);
}

Python

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

↓

def code(u, v, t1):
	return (v / (t1 + u)) / ((-2.0 - (u / t1)) + 1.0)

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)
	return Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-2.0 - Float64(u / t1)) + 1.0))
end

MATLAB

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

↓

function tmp = code(u, v, t1)
	tmp = (v / (t1 + u)) / ((-2.0 - (u / t1)) + 1.0);
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_] := N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.4

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

2. Simplified 1.4

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

   [Start] 18.4	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   *-commutative [=>] 18.4	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   associate-/l* [=>] 16.0	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
   associate-*r/ [<=] 3.5	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
   associate-/r* [=>] 1.4	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}}} \]
   neg-mul-1 [=>] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-1 \cdot t1}}} \]
   associate-/l/ [<=] 1.4	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
   metadata-eval [<=] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 - 1}}} \]
   mul0-lft [<=] 8.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]
   associate-*r/ [=>] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 \cdot \left(t1 + u\right)}{t1}} - 1}} \]
   mul0-lft [=>] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{0}}{t1} - 1}} \]
   *-inverses [<=] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{0}{t1} - \color{blue}{\frac{t1}{t1}}}} \]
   div-sub [<=] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 - t1}{t1}}}} \]
   neg-sub0 [<=] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-t1}}{t1}}} \]
   neg-mul-1 [=>] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-1 \cdot t1}}{t1}}} \]
   *-commutative [=>] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{t1 \cdot -1}}{t1}}} \]
   associate-/l* [=>] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{t1}{\frac{t1}{-1}}}}} \]
   associate-/l* [<=] 1.4	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1} \cdot \frac{t1}{-1}}{t1}}} \]
   *-commutative [=>] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{-1} \cdot \frac{t1 + u}{t1}}}{t1}} \]
   times-frac [<=] 16.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
   neg-mul-1 [<=] 16.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
   associate-/l* [=>] 1.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]

3. Applied egg-rr 1.4

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

4. Applied egg-rr 1.4

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

5. Final simplification 1.4

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

Alternatives

Alternative 1
Error	14.0
Cost	905

\[\begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -2.5 \cdot 10^{-11} \lor \neg \left(t1 \leq 1.05 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{t_1}{-1}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 2
Error	14.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3.8 \cdot 10^{-14} \lor \neg \left(t1 \leq 1.4 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \end{array} \]

Alternative 3
Error	14.7
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -6.6 \cdot 10^{-14} \lor \neg \left(t1 \leq 1.05 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 4
Error	14.5
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -4.2 \cdot 10^{-14} \lor \neg \left(t1 \leq 7.6 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 5
Error	14.0
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -6.5 \cdot 10^{-14} \lor \neg \left(t1 \leq 6.8 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 6
Error	1.2
Cost	768

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

Alternative 7
Error	22.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+138} \lor \neg \left(u \leq 35000000000\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8
Error	22.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -2.55 \cdot 10^{+138}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 33500000000:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 9
Error	22.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+141}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 35000000000:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 10
Error	3.5
Cost	704

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

Alternative 11
Error	1.4
Cost	704

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

Alternative 12
Error	27.9
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -4.5 \cdot 10^{+140} \lor \neg \left(u \leq 2.05 \cdot 10^{+226}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 13
Error	30.7
Cost	256

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

Reproduce

herbie shell --seed 2023067 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))