Rosa's DopplerBench

Average Accuracy: 71.6% → 97.7%

Time: 12.6s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

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)) / (-1.0 - (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 = (-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)) / ((-1.0d0) - (u / t1))
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)) / (-1.0 - (u / t1));
}

Python

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

↓

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

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(-1.0 - Float64(u / t1)))
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)) / (-1.0 - (u / t1));
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[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

2. Simplified 97.7%

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

   [Start] 71.6	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   *-commutative [=>] 71.6	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 97.8	\[ \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]
   neg-mul-1 [=>] 97.8	\[ \frac{v}{t1 + u} \cdot \frac{\color{blue}{-1 \cdot t1}}{t1 + u} \]
   associate-/l* [=>] 97.7	\[ \frac{v}{t1 + u} \cdot \color{blue}{\frac{-1}{\frac{t1 + u}{t1}}} \]
   associate-*r/ [=>] 97.7	\[ \color{blue}{\frac{\frac{v}{t1 + u} \cdot -1}{\frac{t1 + u}{t1}}} \]
   associate-/l* [=>] 97.7	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
   associate-/l/ [=>] 97.7	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{t1 + u}{-1 \cdot t1}}} \]
   neg-mul-1 [<=] 97.7	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-t1}}} \]
   *-lft-identity [<=] 97.7	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{1 \cdot \frac{t1 + u}{-t1}}} \]
   metadata-eval [<=] 97.7	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{t1 + u}{-t1}} \]
   times-frac [<=] 97.7	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1 \cdot \left(t1 + u\right)}{-1 \cdot \left(-t1\right)}}} \]
   neg-mul-1 [<=] 97.7	\[ \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{-\left(-t1\right)}}} \]
   remove-double-neg [=>] 97.7	\[ \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{t1}}} \]
   neg-mul-1 [<=] 97.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{-\left(t1 + u\right)}}{t1}} \]
   sub0-neg [<=] 97.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]
   associate--r+ [=>] 97.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]
   neg-sub0 [<=] 97.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(-t1\right)} - u}{t1}} \]
   div-sub [=>] 97.7	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-t1}{t1} - \frac{u}{t1}}} \]
   distribute-frac-neg [=>] 97.7	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\left(-\frac{t1}{t1}\right)} - \frac{u}{t1}} \]
   *-inverses [=>] 97.7	\[ \frac{\frac{v}{t1 + u}}{\left(-\color{blue}{1}\right) - \frac{u}{t1}} \]
   metadata-eval [=>] 97.7	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{-1} - \frac{u}{t1}} \]

3. Final simplification 97.7%

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

Alternatives

Alternative 1
Accuracy	76.6%
Cost	1300

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -4 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 3.25 \cdot 10^{-131}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 12500:\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq 1.65 \cdot 10^{+161}:\\ \;\;\;\;\frac{-t1}{\frac{t1 + u}{\frac{v}{t1}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	76.8%
Cost	1300

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.45 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 5 \cdot 10^{-131}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 9.5:\\ \;\;\;\;\frac{-t1}{\frac{t1 + u}{\frac{v}{u}}}\\ \mathbf{elif}\;t1 \leq 1.52 \cdot 10^{+161}:\\ \;\;\;\;\frac{-t1}{\frac{t1 + u}{\frac{v}{t1}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	76.0%
Cost	777

\[\begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+54} \lor \neg \left(u \leq 4.2 \cdot 10^{+17}\right):\\ \;\;\;\;t1 \cdot \frac{-\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 4
Accuracy	77.3%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -6.6 \cdot 10^{-83} \lor \neg \left(t1 \leq 5 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \left(-\frac{v}{u}\right)\\ \end{array} \]

Alternative 5
Accuracy	77.5%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -6.5 \cdot 10^{-84} \lor \neg \left(t1 \leq 3.15 \cdot 10^{-134}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 6
Accuracy	77.7%
Cost	776

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 5 \cdot 10^{-131}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 7
Accuracy	66.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{+112} \lor \neg \left(u \leq 4.5 \cdot 10^{+86}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 8
Accuracy	66.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -1.85 \cdot 10^{+55} \lor \neg \left(u \leq 3.7 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 9
Accuracy	67.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -1.95 \cdot 10^{+55} \lor \neg \left(u \leq 3.1 \cdot 10^{+86}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 10
Accuracy	67.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+181} \lor \neg \left(u \leq 1.4 \cdot 10^{+117}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 11
Accuracy	94.7%
Cost	704

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

Alternative 12
Accuracy	56.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -1.65 \cdot 10^{+182} \lor \neg \left(u \leq 5.2 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 13
Accuracy	57.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -1.8 \cdot 10^{+181} \lor \neg \left(u \leq 1.95 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 14
Accuracy	56.9%
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -4.5 \cdot 10^{+181} \lor \neg \left(u \leq 1.12 \cdot 10^{+104}\right):\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 15
Accuracy	52.7%
Cost	256

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

Reproduce

herbie shell --seed 2023146 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))