Rosa's DopplerBench

Average Accuracy: 71.5% → 97.8%

Time: 13.3s

Precision: binary64

Cost: 768

Math

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

↓

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

FPCore

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

↓

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

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 (-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 * 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 = (-t1 / (t1 + u)) * (v / (t1 + u))
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 (-t1 / (t1 + u)) * (v / (t1 + u));
}

Python

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

↓

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

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(Float64(-t1) / Float64(t1 + u)) * Float64(v / Float64(t1 + u)))
end

MATLAB

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

↓

function tmp = code(u, v, t1)
	tmp = (-t1 / (t1 + u)) * (v / (t1 + u));
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[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

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

2. Simplified 97.8%

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

   [Start] 71.5	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 97.8	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 97.8%

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

Alternatives

Alternative 1
Accuracy	76.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{-61} \lor \neg \left(u \leq 2.95 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 2
Accuracy	78.4%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -5.6 \cdot 10^{-65} \lor \neg \left(t1 \leq 6.5 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 3
Accuracy	76.0%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{-64} \lor \neg \left(t1 \leq 2.9 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 4
Accuracy	74.8%
Cost	777

\[\begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{-61} \lor \neg \left(u \leq 6 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 5
Accuracy	76.1%
Cost	777

\[\begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{-61} \lor \neg \left(u \leq 2.2 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 6
Accuracy	74.7%
Cost	776

\[\begin{array}{l} \mathbf{if}\;u \leq -2.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq 6 \cdot 10^{-79}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \]

Alternative 7
Accuracy	65.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+63} \lor \neg \left(u \leq 4.2 \cdot 10^{+58}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8
Accuracy	66.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.85 \cdot 10^{-156} \lor \neg \left(t1 \leq 3.35 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \]

Alternative 9
Accuracy	66.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t1 \leq -8.4 \cdot 10^{-156} \lor \neg \left(t1 \leq 7.2 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{u}\\ \end{array} \]

Alternative 10
Accuracy	65.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+63}:\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 11
Accuracy	66.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+63}:\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 12
Accuracy	94.7%
Cost	704

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

Alternative 13
Accuracy	97.7%
Cost	704

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

Alternative 14
Accuracy	56.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -2.5 \cdot 10^{+154} \lor \neg \left(u \leq 9 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 15
Accuracy	57.1%
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+154} \lor \neg \left(u \leq 1.75 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 16
Accuracy	56.9%
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+154} \lor \neg \left(u \leq 2.4 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 17
Accuracy	52.3%
Cost	256

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

Reproduce

herbie shell --seed 2023122 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))