Rosa's DopplerBench

Average Error: 17.7 → 1.3

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 17.7

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

2. Simplified 1.3

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

   [Start] 17.7	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 1.3	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 1.3

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

Alternatives

Alternative 1
Error	19.3
Cost	1042

\[\begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{+66} \lor \neg \left(u \leq -2.4 \cdot 10^{-27} \lor \neg \left(u \leq -1.85 \cdot 10^{-58}\right) \land u \leq 7.2 \cdot 10^{+128}\right):\\ \;\;\;\;\frac{-t1}{\frac{u \cdot u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \end{array} \]

Alternative 2
Error	19.0
Cost	1041

\[\begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{+66}:\\ \;\;\;\;\frac{-t1}{\frac{u \cdot u}{v}}\\ \mathbf{elif}\;u \leq -2.7 \cdot 10^{-25} \lor \neg \left(u \leq -1.22 \cdot 10^{-67}\right) \land u \leq 1.75 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 3
Error	19.0
Cost	1041

\[\begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{+66}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{elif}\;u \leq -2.6 \cdot 10^{-28} \lor \neg \left(u \leq -1.22 \cdot 10^{-67}\right) \land u \leq 1.16 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 4
Error	16.7
Cost	1040

\[\begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{-1}\\ t_2 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -1.05 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -2.1 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -1.75 \cdot 10^{-69}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{-u}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	16.7
Cost	1040

\[\begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{-1}\\ t_2 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -1.5 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -1.35 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -1.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 1.65 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	16.6
Cost	1040

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ t_2 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -3 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -1.12 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -4.7 \cdot 10^{-68}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	16.4
Cost	1040

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ t_2 := \frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{if}\;u \leq -1.8 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -5.5 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -1.25 \cdot 10^{-105}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 1.12 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 8
Error	3.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+139}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \end{array} \]

Alternative 9
Error	21.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{+67} \lor \neg \left(u \leq 8.2 \cdot 10^{+132}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10
Error	21.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -1.95 \cdot 10^{+68}:\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+136}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 11
Error	20.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+71}:\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 12
Error	1.5
Cost	704

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

Alternative 13
Error	1.3
Cost	704

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

Alternative 14
Error	27.6
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+173} \lor \neg \left(u \leq 1.08 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 15
Error	30.7
Cost	256

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

Alternative 16
Error	54.7
Cost	192

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

Reproduce

herbie shell --seed 2023060 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))